rsin A (should all be same)

Percentage Accurate: 76.4% → 99.4%

Time: 17.7s

Alternatives: 18

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: 76.4% 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: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot \sin a\\ \frac{\sin b \cdot r}{-2 \cdot t_0 + \left(t_0 + \cos a \cdot \cos b\right)} \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (sin a))))
   (/ (* (sin b) r) (+ (* -2.0 t_0) (+ t_0 (* (cos a) (cos b)))))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * sin(a);
	return (sin(b) * r) / ((-2.0 * t_0) + (t_0 + (cos(a) * cos(b))));
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    t_0 = sin(b) * sin(a)
    code = (sin(b) * r) / (((-2.0d0) * t_0) + (t_0 + (cos(a) * cos(b))))
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.sin(b) * Math.sin(a);
	return (Math.sin(b) * r) / ((-2.0 * t_0) + (t_0 + (Math.cos(a) * Math.cos(b))));
}

Python

def code(r, a, b):
	t_0 = math.sin(b) * math.sin(a)
	return (math.sin(b) * r) / ((-2.0 * t_0) + (t_0 + (math.cos(a) * math.cos(b))))

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * sin(a))
	return Float64(Float64(sin(b) * r) / Float64(Float64(-2.0 * t_0) + Float64(t_0 + Float64(cos(a) * cos(b)))))
end

MATLAB

function tmp = code(r, a, b)
	t_0 = sin(b) * sin(a);
	tmp = (sin(b) * r) / ((-2.0 * t_0) + (t_0 + (cos(a) * cos(b))));
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[(N[(-2.0 * t$95$0), $MachinePrecision] + N[(t$95$0 + N[(N[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 70.2%

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

   1. +-commutative 70.2%

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

3. Simplified 70.2%

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

   1. cos-sum 99.4%

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

   2. *-un-lft-identity 99.4%

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

   3. *-un-lft-identity 99.4%

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

   4. prod-diff 99.4%

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

5. Applied egg-rr 99.4%

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

   1. *-rgt-identity 99.4%

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

   2. *-commutative 99.4%

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

   3. distribute-lft-neg-in 99.4%

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

   4. *-commutative 99.4%

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

   5. fma-udef 99.4%

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

   6. *-rgt-identity 99.4%

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

   7. distribute-lft-neg-in 99.4%

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

   8. *-rgt-identity 99.4%

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

   9. fma-udef 99.4%

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

   10. *-commutative 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in r around 0 99.4%

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

9. Final simplification 99.4%

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

Alternative 2: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[r_, a_, b_] := N[(N[(N[Sin[b], $MachinePrecision] * r), $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 70.2%

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

   1. +-commutative 70.2%

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

3. Simplified 70.2%

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

   1. cos-sum 99.4%

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

   2. cancel-sign-sub-inv 99.4%

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

   3. fma-def 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.2%

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

   1. associate-*r/ 70.1%

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

   2. *-commutative 70.1%

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

   3. +-commutative 70.1%

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

3. Simplified 70.1%

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

   1. cos-sum 99.4%

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

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%

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

Alternative 4: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.2%

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

   1. +-commutative 70.2%

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

3. Simplified 70.2%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 5: 77.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\sin b \cdot r}{-2 \cdot \left(\sin b \cdot \sin a\right) + \cos \left(a - b\right)} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ (* (sin b) r) (+ (* -2.0 (* (sin b) (sin a))) (cos (- a b)))))

C

double code(double r, double a, double b) {
	return (sin(b) * r) / ((-2.0 * (sin(b) * sin(a))) + 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 = (sin(b) * r) / (((-2.0d0) * (sin(b) * sin(a))) + cos((a - b)))
end function

Java

public static double code(double r, double a, double b) {
	return (Math.sin(b) * r) / ((-2.0 * (Math.sin(b) * Math.sin(a))) + Math.cos((a - b)));
}

Python

def code(r, a, b):
	return (math.sin(b) * r) / ((-2.0 * (math.sin(b) * math.sin(a))) + math.cos((a - b)))

Julia

function code(r, a, b)
	return Float64(Float64(sin(b) * r) / Float64(Float64(-2.0 * Float64(sin(b) * sin(a))) + cos(Float64(a - b))))
end

MATLAB

function tmp = code(r, a, b)
	tmp = (sin(b) * r) / ((-2.0 * (sin(b) * sin(a))) + cos((a - b)));
end

Wolfram

code[r_, a_, b_] := N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[(N[(-2.0 * N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Cos[N[(a - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.2%

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

   1. +-commutative 70.2%

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

3. Simplified 70.2%

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

   1. cos-sum 99.4%

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

   2. *-un-lft-identity 99.4%

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

   3. *-un-lft-identity 99.4%

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

   4. prod-diff 99.4%

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

5. Applied egg-rr 99.4%

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

   1. *-rgt-identity 99.4%

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

   2. *-commutative 99.4%

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

   3. distribute-lft-neg-in 99.4%

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

   4. *-commutative 99.4%

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

   5. fma-udef 99.4%

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

   6. *-rgt-identity 99.4%

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

   7. distribute-lft-neg-in 99.4%

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

   8. *-rgt-identity 99.4%

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

   9. fma-udef 99.4%

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

   10. *-commutative 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in r around 0 99.4%

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

   1. expm1-log1p-u 99.3%

      \[\leadsto \frac{\sin b \cdot r}{-2 \cdot \left(\sin a \cdot \sin b\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b + \cos a \cdot \cos b\right)\right)}} \]

   2. expm1-udef 99.0%

      \[\leadsto \frac{\sin b \cdot r}{-2 \cdot \left(\sin a \cdot \sin b\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\sin a \cdot \sin b + \cos a \cdot \cos b\right)} - 1\right)}} \]

   3. +-commutative 99.0%

      \[\leadsto \frac{\sin b \cdot r}{-2 \cdot \left(\sin a \cdot \sin b\right) + \left(e^{\mathsf{log1p}\left(\color{blue}{\cos a \cdot \cos b + \sin a \cdot \sin b}\right)} - 1\right)} \]

   4. cos-diff 71.0%

      \[\leadsto \frac{\sin b \cdot r}{-2 \cdot \left(\sin a \cdot \sin b\right) + \left(e^{\mathsf{log1p}\left(\color{blue}{\cos \left(a - b\right)}\right)} - 1\right)} \]

10. Applied egg-rr 71.0%

    \[\leadsto \frac{\sin b \cdot r}{-2 \cdot \left(\sin a \cdot \sin b\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(a - b\right)\right)} - 1\right)}} \]
11. Step-by-step derivation

    1. expm1-def 71.3%

       \[\leadsto \frac{\sin b \cdot r}{-2 \cdot \left(\sin a \cdot \sin b\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(a - b\right)\right)\right)}} \]

    2. expm1-log1p 71.3%

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

12. Simplified 71.3%

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

13. Final simplification 71.3%

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

Alternative 6: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0036 \lor \neg \left(b \leq 0.00245\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.0036) (not (<= b 0.00245)))
   (* (sin b) (/ r (cos b)))
   (/ (* b r) (cos (+ b a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.0036) || !(b <= 0.00245)) {
		tmp = sin(b) * (r / cos(b));
	} else {
		tmp = (b * r) / cos((b + a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-0.0036d0)) .or. (.not. (b <= 0.00245d0))) then
        tmp = sin(b) * (r / cos(b))
    else
        tmp = (b * r) / cos((b + a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.0036) || !(b <= 0.00245)) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else {
		tmp = (b * r) / Math.cos((b + a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.0036) or not (b <= 0.00245):
		tmp = math.sin(b) * (r / math.cos(b))
	else:
		tmp = (b * r) / math.cos((b + a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.0036) || !(b <= 0.00245))
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	else
		tmp = Float64(Float64(b * r) / cos(Float64(b + a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.0036) || ~((b <= 0.00245)))
		tmp = sin(b) * (r / cos(b));
	else
		tmp = (b * r) / cos((b + a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.0036], N[Not[LessEqual[b, 0.00245]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * r), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.0035999999999999999 or 0.0024499999999999999 < b

   1. Initial program 47.4%

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

      1. +-commutative 47.4%

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

   3. Simplified 47.4%

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

      1. cos-sum 99.2%

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

      2. cancel-sign-sub-inv 99.2%

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

      3. fma-def 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in a around 0 47.6%

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

      1. associate-*r/ 47.7%

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

   8. Simplified 47.7%

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

   if -0.0035999999999999999 < b < 0.0024499999999999999

   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%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0036 \lor \neg \left(b \leq 0.00245\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 7: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0062 \lor \neg \left(a \leq 0.0027\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.0062) (not (<= a 0.0027)))
   (* r (/ (sin b) (cos a)))
   (* r (/ 1.0 (- (/ 1.0 (tan b)) a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.0062) || !(a <= 0.0027)) {
		tmp = r * (sin(b) / cos(a));
	} else {
		tmp = r * (1.0 / ((1.0 / tan(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 ((a <= (-0.0062d0)) .or. (.not. (a <= 0.0027d0))) then
        tmp = r * (sin(b) / cos(a))
    else
        tmp = r * (1.0d0 / ((1.0d0 / tan(b)) - a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.0062) || !(a <= 0.0027)) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else {
		tmp = r * (1.0 / ((1.0 / Math.tan(b)) - a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -0.0062) or not (a <= 0.0027):
		tmp = r * (math.sin(b) / math.cos(a))
	else:
		tmp = r * (1.0 / ((1.0 / math.tan(b)) - a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -0.0062) || !(a <= 0.0027))
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	else
		tmp = Float64(r * Float64(1.0 / Float64(Float64(1.0 / tan(b)) - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -0.0062) || ~((a <= 0.0027)))
		tmp = r * (sin(b) / cos(a));
	else
		tmp = r * (1.0 / ((1.0 / tan(b)) - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -0.0062], N[Not[LessEqual[a, 0.0027]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(1.0 / N[(N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.00619999999999999978 or 0.0027000000000000001 < a

   1. Initial program 48.4%

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

      1. associate-*r/ 48.3%

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

      2. *-commutative 48.3%

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

      3. +-commutative 48.3%

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

   3. Simplified 48.3%

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

   4. Taylor expanded in b around 0 48.1%

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

   if -0.00619999999999999978 < a < 0.0027000000000000001

   1. Initial program 98.2%

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

      1. associate-/l* 98.1%

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

      2. +-commutative 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in a around 0 99.3%

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

      1. mul-1-neg 99.3%

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

      2. unsub-neg 99.3%

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

   6. Simplified 99.3%

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

      1. div-inv 99.2%

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

      2. clear-num 99.2%

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

      3. quot-tan 99.4%

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

   8. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{r \cdot \frac{1}{\frac{1}{\tan b} - a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0062 \lor \neg \left(a \leq 0.0027\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \end{array} \]

Alternative 8: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.056 \lor \neg \left(a \leq 0.00058\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -0.056) (not (<= a 0.00058)))
   (/ r (/ (cos a) (sin b)))
   (* r (/ 1.0 (- (/ 1.0 (tan b)) a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.056) || !(a <= 0.00058)) {
		tmp = r / (cos(a) / sin(b));
	} else {
		tmp = r * (1.0 / ((1.0 / tan(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 ((a <= (-0.056d0)) .or. (.not. (a <= 0.00058d0))) then
        tmp = r / (cos(a) / sin(b))
    else
        tmp = r * (1.0d0 / ((1.0d0 / tan(b)) - a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -0.056) || !(a <= 0.00058)) {
		tmp = r / (Math.cos(a) / Math.sin(b));
	} else {
		tmp = r * (1.0 / ((1.0 / Math.tan(b)) - a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -0.056) or not (a <= 0.00058):
		tmp = r / (math.cos(a) / math.sin(b))
	else:
		tmp = r * (1.0 / ((1.0 / math.tan(b)) - a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -0.056) || !(a <= 0.00058))
		tmp = Float64(r / Float64(cos(a) / sin(b)));
	else
		tmp = Float64(r * Float64(1.0 / Float64(Float64(1.0 / tan(b)) - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -0.056) || ~((a <= 0.00058)))
		tmp = r / (cos(a) / sin(b));
	else
		tmp = r * (1.0 / ((1.0 / tan(b)) - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -0.056], N[Not[LessEqual[a, 0.00058]], $MachinePrecision]], N[(r / N[(N[Cos[a], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(1.0 / N[(N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0560000000000000012 or 5.8e-4 < a

   1. Initial program 48.4%

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

      1. associate-/l* 48.4%

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

      2. +-commutative 48.4%

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

   3. Simplified 48.4%

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

   4. Taylor expanded in b around 0 48.2%

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

   if -0.0560000000000000012 < a < 5.8e-4

   1. Initial program 98.2%

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

      1. associate-/l* 98.1%

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

      2. +-commutative 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in a around 0 99.3%

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

      1. mul-1-neg 99.3%

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

      2. unsub-neg 99.3%

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

   6. Simplified 99.3%

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

      1. div-inv 99.2%

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

      2. clear-num 99.2%

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

      3. quot-tan 99.4%

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

   8. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{r \cdot \frac{1}{\frac{1}{\tan b} - a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.056 \lor \neg \left(a \leq 0.00058\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \end{array} \]

Alternative 9: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0045:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \mathbf{elif}\;a \leq 0.03:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \mathsf{log1p}\left(\frac{b}{\cos a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.0045)
   (/ (* b r) (cos (+ b a)))
   (if (<= a 0.03)
     (* r (/ 1.0 (- (/ 1.0 (tan b)) a)))
     (* r (log1p (/ b (cos a)))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.0045) {
		tmp = (b * r) / cos((b + a));
	} else if (a <= 0.03) {
		tmp = r * (1.0 / ((1.0 / tan(b)) - a));
	} else {
		tmp = r * log1p((b / cos(a)));
	}
	return tmp;
}

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.0045) {
		tmp = (b * r) / Math.cos((b + a));
	} else if (a <= 0.03) {
		tmp = r * (1.0 / ((1.0 / Math.tan(b)) - a));
	} else {
		tmp = r * Math.log1p((b / Math.cos(a)));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -0.0045:
		tmp = (b * r) / math.cos((b + a))
	elif a <= 0.03:
		tmp = r * (1.0 / ((1.0 / math.tan(b)) - a))
	else:
		tmp = r * math.log1p((b / math.cos(a)))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -0.0045)
		tmp = Float64(Float64(b * r) / cos(Float64(b + a)));
	elseif (a <= 0.03)
		tmp = Float64(r * Float64(1.0 / Float64(Float64(1.0 / tan(b)) - a)));
	else
		tmp = Float64(r * log1p(Float64(b / cos(a))));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -0.0045], N[(N[(b * r), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.03], N[(r * N[(1.0 / N[(N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[Log[1 + N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.00449999999999999966

   1. Initial program 46.5%

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

      1. +-commutative 46.5%

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

   3. Simplified 46.5%

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

   4. Taylor expanded in b around 0 43.3%

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

   if -0.00449999999999999966 < a < 0.029999999999999999

   1. Initial program 98.2%

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

      1. associate-/l* 98.1%

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

      2. +-commutative 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in a around 0 99.3%

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

      1. mul-1-neg 99.3%

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

      2. unsub-neg 99.3%

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

   6. Simplified 99.3%

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

      1. div-inv 99.2%

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

      2. clear-num 99.2%

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

      3. quot-tan 99.4%

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

   8. Applied egg-rr 99.4%

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

   if 0.029999999999999999 < a

   1. Initial program 50.3%

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

      1. associate-*r/ 50.3%

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

      2. *-commutative 50.3%

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

      3. +-commutative 50.3%

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

   3. Simplified 50.3%

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

      1. add-log-exp 12.9%

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

   5. Applied egg-rr 12.9%

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

   6. Taylor expanded in b around 0 8.6%

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

      1. *-un-lft-identity 8.6%

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

      2. log-prod 8.6%

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

      3. metadata-eval 8.6%

         \[\leadsto \left(\color{blue}{0} + \log \left(\frac{b}{\cos a} + 1\right)\right) \cdot r \]

      4. +-commutative 8.6%

         \[\leadsto \left(0 + \log \color{blue}{\left(1 + \frac{b}{\cos a}\right)}\right) \cdot r \]

      5. log1p-udef 46.0%

         \[\leadsto \left(0 + \color{blue}{\mathsf{log1p}\left(\frac{b}{\cos a}\right)}\right) \cdot r \]

   8. Applied egg-rr 46.0%

      \[\leadsto \color{blue}{\left(0 + \mathsf{log1p}\left(\frac{b}{\cos a}\right)\right)} \cdot r \]
   9. Step-by-step derivation

      1. +-lft-identity 46.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{b}{\cos a}\right)} \cdot r \]

   10. Simplified 46.0%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{b}{\cos a}\right)} \cdot r \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0045:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \mathbf{elif}\;a \leq 0.03:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \mathsf{log1p}\left(\frac{b}{\cos a}\right)\\ \end{array} \]

Alternative 10: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0028:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{elif}\;a \leq 0.0029:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.0028)
   (/ (sin b) (/ (cos a) r))
   (if (<= a 0.0029)
     (* r (/ 1.0 (- (/ 1.0 (tan b)) a)))
     (/ r (/ (cos a) (sin b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.0028) {
		tmp = sin(b) / (cos(a) / r);
	} else if (a <= 0.0029) {
		tmp = r * (1.0 / ((1.0 / tan(b)) - a));
	} else {
		tmp = r / (cos(a) / 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 (a <= (-0.0028d0)) then
        tmp = sin(b) / (cos(a) / r)
    else if (a <= 0.0029d0) then
        tmp = r * (1.0d0 / ((1.0d0 / tan(b)) - a))
    else
        tmp = r / (cos(a) / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.0028) {
		tmp = Math.sin(b) / (Math.cos(a) / r);
	} else if (a <= 0.0029) {
		tmp = r * (1.0 / ((1.0 / Math.tan(b)) - a));
	} else {
		tmp = r / (Math.cos(a) / Math.sin(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -0.0028:
		tmp = math.sin(b) / (math.cos(a) / r)
	elif a <= 0.0029:
		tmp = r * (1.0 / ((1.0 / math.tan(b)) - a))
	else:
		tmp = r / (math.cos(a) / math.sin(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -0.0028)
		tmp = Float64(sin(b) / Float64(cos(a) / r));
	elseif (a <= 0.0029)
		tmp = Float64(r * Float64(1.0 / Float64(Float64(1.0 / tan(b)) - a)));
	else
		tmp = Float64(r / Float64(cos(a) / sin(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -0.0028)
		tmp = sin(b) / (cos(a) / r);
	elseif (a <= 0.0029)
		tmp = r * (1.0 / ((1.0 / tan(b)) - a));
	else
		tmp = r / (cos(a) / sin(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -0.0028], N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[a], $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.0029], N[(r * N[(1.0 / N[(N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r / N[(N[Cos[a], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.00279999999999999997

   1. Initial program 46.5%

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

      1. associate-*r/ 46.5%

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

      2. *-commutative 46.5%

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

      3. +-commutative 46.5%

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

   3. Simplified 46.5%

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

      1. associate-*l/ 46.5%

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

      2. associate-/l* 46.5%

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

   5. Applied egg-rr 46.5%

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

   6. Taylor expanded in b around 0 46.8%

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

   if -0.00279999999999999997 < a < 0.0029

   1. Initial program 98.2%

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

      1. associate-/l* 98.1%

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

      2. +-commutative 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in a around 0 99.3%

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

      1. mul-1-neg 99.3%

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

      2. unsub-neg 99.3%

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

   6. Simplified 99.3%

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

      1. div-inv 99.2%

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

      2. clear-num 99.2%

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

      3. quot-tan 99.4%

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

   8. Applied egg-rr 99.4%

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

   if 0.0029 < a

   1. Initial program 50.3%

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

      1. associate-/l* 50.4%

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

      2. +-commutative 50.4%

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

   3. Simplified 50.4%

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

   4. Taylor expanded in b around 0 49.6%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0028:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{elif}\;a \leq 0.0029:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \end{array} \]

Alternative 11: 76.4% 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 70.2%

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

   1. associate-/l* 70.1%

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

   2. +-commutative 70.1%

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

3. Simplified 70.1%

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

   1. associate-/r/ 70.2%

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

5. Applied egg-rr 70.2%

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

6. Final simplification 70.2%

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

Alternative 12: 74.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\tan b} - a\\ \mathbf{if}\;b \leq -0.053:\\ \;\;\;\;\frac{r}{t_0}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 (tan b)) a)))
   (if (<= b -0.053)
     (/ r t_0)
     (if (<= b 8.2e-5) (/ (* b r) (cos (+ b a))) (* r (/ 1.0 t_0))))))

C

double code(double r, double a, double b) {
	double t_0 = (1.0 / tan(b)) - a;
	double tmp;
	if (b <= -0.053) {
		tmp = r / t_0;
	} else if (b <= 8.2e-5) {
		tmp = (b * r) / cos((b + a));
	} else {
		tmp = r * (1.0 / t_0);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / tan(b)) - a
    if (b <= (-0.053d0)) then
        tmp = r / t_0
    else if (b <= 8.2d-5) then
        tmp = (b * r) / cos((b + a))
    else
        tmp = r * (1.0d0 / t_0)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (1.0 / Math.tan(b)) - a;
	double tmp;
	if (b <= -0.053) {
		tmp = r / t_0;
	} else if (b <= 8.2e-5) {
		tmp = (b * r) / Math.cos((b + a));
	} else {
		tmp = r * (1.0 / t_0);
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (1.0 / math.tan(b)) - a
	tmp = 0
	if b <= -0.053:
		tmp = r / t_0
	elif b <= 8.2e-5:
		tmp = (b * r) / math.cos((b + a))
	else:
		tmp = r * (1.0 / t_0)
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(1.0 / tan(b)) - a)
	tmp = 0.0
	if (b <= -0.053)
		tmp = Float64(r / t_0);
	elseif (b <= 8.2e-5)
		tmp = Float64(Float64(b * r) / cos(Float64(b + a)));
	else
		tmp = Float64(r * Float64(1.0 / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (1.0 / tan(b)) - a;
	tmp = 0.0;
	if (b <= -0.053)
		tmp = r / t_0;
	elseif (b <= 8.2e-5)
		tmp = (b * r) / cos((b + a));
	else
		tmp = r * (1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]}, If[LessEqual[b, -0.053], N[(r / t$95$0), $MachinePrecision], If[LessEqual[b, 8.2e-5], N[(N[(b * r), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(r * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.0529999999999999985

   1. Initial program 50.5%

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

      1. associate-/l* 50.5%

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

      2. +-commutative 50.5%

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

   3. Simplified 50.5%

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

   4. Taylor expanded in a around 0 46.9%

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

      1. mul-1-neg 46.9%

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

      2. unsub-neg 46.9%

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

   6. Simplified 46.9%

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

      1. expm1-log1p-u 33.5%

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

      2. expm1-udef 33.3%

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

      3. clear-num 33.3%

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

      4. quot-tan 33.3%

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

   8. Applied egg-rr 33.3%

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

      1. expm1-def 33.6%

         \[\leadsto \frac{r}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\tan b}\right)\right)} - a} \]

      2. expm1-log1p 47.0%

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

   10. Simplified 47.0%

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

   if -0.0529999999999999985 < b < 8.20000000000000009e-5

   1. Initial program 97.9%

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

      1. +-commutative 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in b around 0 97.9%

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

   if 8.20000000000000009e-5 < b

   1. Initial program 45.0%

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

      1. associate-/l* 45.1%

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

      2. +-commutative 45.1%

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

   3. Simplified 45.1%

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

   4. Taylor expanded in a around 0 42.6%

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

      1. mul-1-neg 42.6%

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

      2. unsub-neg 42.6%

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

   6. Simplified 42.6%

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

      1. div-inv 42.5%

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

      2. clear-num 42.6%

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

      3. quot-tan 42.6%

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

   8. Applied egg-rr 42.6%

      \[\leadsto \color{blue}{r \cdot \frac{1}{\frac{1}{\tan b} - a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.053:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b} - a}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \end{array} \]

Alternative 13: 74.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.053 \lor \neg \left(b \leq 0.00023\right):\\ \;\;\;\;\frac{r}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.053) (not (<= b 0.00023)))
   (/ r (- (/ 1.0 (tan b)) a))
   (/ (* b r) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.053) || !(b <= 0.00023)) {
		tmp = r / ((1.0 / tan(b)) - 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 <= (-0.053d0)) .or. (.not. (b <= 0.00023d0))) then
        tmp = r / ((1.0d0 / tan(b)) - 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 <= -0.053) || !(b <= 0.00023)) {
		tmp = r / ((1.0 / Math.tan(b)) - a);
	} else {
		tmp = (b * r) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.053) or not (b <= 0.00023):
		tmp = r / ((1.0 / math.tan(b)) - a)
	else:
		tmp = (b * r) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.053) || !(b <= 0.00023))
		tmp = Float64(r / Float64(Float64(1.0 / tan(b)) - a));
	else
		tmp = Float64(Float64(b * r) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.053) || ~((b <= 0.00023)))
		tmp = r / ((1.0 / tan(b)) - a);
	else
		tmp = (b * r) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.053], N[Not[LessEqual[b, 0.00023]], $MachinePrecision]], N[(r / N[(N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(N[(b * r), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.0529999999999999985 or 2.3000000000000001e-4 < b

   1. Initial program 47.5%

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

      1. associate-/l* 47.6%

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

      2. +-commutative 47.6%

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

   3. Simplified 47.6%

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

   4. Taylor expanded in a around 0 44.6%

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

      1. mul-1-neg 44.6%

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

      2. unsub-neg 44.6%

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

   6. Simplified 44.6%

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

      1. expm1-log1p-u 32.3%

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

      2. expm1-udef 32.1%

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

      3. clear-num 32.0%

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

      4. quot-tan 32.0%

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

   8. Applied egg-rr 32.0%

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

      1. expm1-def 32.3%

         \[\leadsto \frac{r}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\tan b}\right)\right)} - a} \]

      2. expm1-log1p 44.6%

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

   10. Simplified 44.6%

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

   if -0.0529999999999999985 < b < 2.3000000000000001e-4

   1. Initial program 97.9%

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

      1. associate-*r/ 97.9%

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

      2. *-commutative 97.9%

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

      3. +-commutative 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in b around 0 97.5%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.053 \lor \neg \left(b \leq 0.00023\right):\\ \;\;\;\;\frac{r}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \end{array} \]

Alternative 14: 74.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.053 \lor \neg \left(b \leq 0.00028\right):\\ \;\;\;\;\frac{r}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.053) (not (<= b 0.00028)))
   (/ r (- (/ 1.0 (tan b)) a))
   (/ (* b r) (cos (+ b a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.053) || !(b <= 0.00028)) {
		tmp = r / ((1.0 / tan(b)) - a);
	} else {
		tmp = (b * r) / cos((b + a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-0.053d0)) .or. (.not. (b <= 0.00028d0))) then
        tmp = r / ((1.0d0 / tan(b)) - a)
    else
        tmp = (b * r) / cos((b + a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.053) || !(b <= 0.00028)) {
		tmp = r / ((1.0 / Math.tan(b)) - a);
	} else {
		tmp = (b * r) / Math.cos((b + a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.053) or not (b <= 0.00028):
		tmp = r / ((1.0 / math.tan(b)) - a)
	else:
		tmp = (b * r) / math.cos((b + a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.053) || !(b <= 0.00028))
		tmp = Float64(r / Float64(Float64(1.0 / tan(b)) - a));
	else
		tmp = Float64(Float64(b * r) / cos(Float64(b + a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.053) || ~((b <= 0.00028)))
		tmp = r / ((1.0 / tan(b)) - a);
	else
		tmp = (b * r) / cos((b + a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.053], N[Not[LessEqual[b, 0.00028]], $MachinePrecision]], N[(r / N[(N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(N[(b * r), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.0529999999999999985 or 2.7999999999999998e-4 < b

   1. Initial program 47.5%

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

      1. associate-/l* 47.6%

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

      2. +-commutative 47.6%

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

   3. Simplified 47.6%

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

   4. Taylor expanded in a around 0 44.6%

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

      1. mul-1-neg 44.6%

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

      2. unsub-neg 44.6%

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

   6. Simplified 44.6%

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

      1. expm1-log1p-u 32.3%

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

      2. expm1-udef 32.1%

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

      3. clear-num 32.0%

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

      4. quot-tan 32.0%

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

   8. Applied egg-rr 32.0%

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

      1. expm1-def 32.3%

         \[\leadsto \frac{r}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\tan b}\right)\right)} - a} \]

      2. expm1-log1p 44.6%

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

   10. Simplified 44.6%

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

   if -0.0529999999999999985 < b < 2.7999999999999998e-4

   1. Initial program 97.9%

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

      1. +-commutative 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in b around 0 97.9%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.053 \lor \neg \left(b \leq 0.00028\right):\\ \;\;\;\;\frac{r}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 15: 50.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.2%

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

   1. associate-*r/ 70.1%

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

   2. *-commutative 70.1%

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

   3. +-commutative 70.1%

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

3. Simplified 70.1%

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

4. Taylor expanded in b around 0 46.0%

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

5. Final simplification 46.0%

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

Alternative 16: 50.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.2%

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

   1. associate-*r/ 70.1%

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

   2. *-commutative 70.1%

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

   3. +-commutative 70.1%

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

3. Simplified 70.1%

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

4. Taylor expanded in b around 0 46.1%

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

5. Final simplification 46.1%

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

Alternative 17: 34.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ r \cdot \mathsf{log1p}\left(b\right) \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* r (log1p b)))

C

double code(double r, double a, double b) {
	return r * log1p(b);
}

Java

public static double code(double r, double a, double b) {
	return r * Math.log1p(b);
}

Python

def code(r, a, b):
	return r * math.log1p(b)

Julia

function code(r, a, b)
	return Float64(r * log1p(b))
end

Wolfram

code[r_, a_, b_] := N[(r * N[Log[1 + b], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.2%

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

   1. associate-*r/ 70.1%

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

   2. *-commutative 70.1%

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

   3. +-commutative 70.1%

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

3. Simplified 70.1%

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

   1. add-log-exp 37.4%

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

5. Applied egg-rr 37.4%

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

6. Taylor expanded in b around 0 13.6%

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

7. Taylor expanded in a around 0 13.2%

   \[\leadsto \color{blue}{\log \left(1 + b\right)} \cdot r \]
8. Step-by-step derivation

   1. log1p-def 29.3%

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

9. Simplified 29.3%

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

10. Final simplification 29.3%

    \[\leadsto r \cdot \mathsf{log1p}\left(b\right) \]

Alternative 18: 34.1% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.2%

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

   1. associate-*r/ 70.1%

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

   2. *-commutative 70.1%

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

   3. +-commutative 70.1%

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

3. Simplified 70.1%

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

4. Taylor expanded in b around 0 46.0%

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

5. Taylor expanded in a around 0 29.3%

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

6. Final simplification 29.3%

   \[\leadsto b \cdot r \]

Reproduce

herbie shell --seed 2023199 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))