rsin B (should all be same)

Percentage Accurate: 77.3% → 99.5%

Time: 16.0s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

   1. associate-*r/ 79.8%

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

   2. /-rgt-identity 79.8%

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

   3. metadata-eval 79.8%

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

   4. associate-/l/ 79.8%

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

   5. associate-*l/ 79.9%

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

   6. associate-/l/ 79.9%

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

   7. metadata-eval 79.9%

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

   8. metadata-eval 79.9%

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

   9. metadata-eval 79.9%

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

   10. /-rgt-identity 79.9%

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

   11. +-commutative 79.9%

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

3. Simplified 79.9%

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

   1. cos-sum 99.5%

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

   2. *-un-lft-identity 99.5%

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

   3. prod-diff 99.5%

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

6. Applied egg-rr 99.5%

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

   1. *-rgt-identity 99.5%

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

   2. fma-neg 99.5%

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

   3. *-commutative 99.5%

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

   4. *-commutative 99.5%

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

   5. fma-udef 99.5%

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

   6. *-rgt-identity 99.5%

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

   7. distribute-lft-neg-in 99.5%

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

   8. *-rgt-identity 99.5%

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

   9. fma-udef 99.5%

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

   10. *-commutative 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = r * (sin(b) / ((cos(a) * cos(b)) - (sin(a) * sin(b))))
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(a) * Math.sin(b))));
}

Python

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

Julia

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

MATLAB

function tmp = code(r, a, b)
	tmp = r * (sin(b) / ((cos(a) * cos(b)) - (sin(a) * sin(b))));
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[a], $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.8%

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

   1. +-commutative 79.8%

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

3. Simplified 79.8%

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

   1. cos-sum 99.5%

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

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. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = sin(b) * (r / ((cos(a) * cos(b)) - (sin(a) * sin(b))))
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(a) * Math.sin(b))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

   1. associate-*r/ 79.8%

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

   2. /-rgt-identity 79.8%

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

   3. metadata-eval 79.8%

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

   4. associate-/l/ 79.8%

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

   5. associate-*l/ 79.9%

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

   6. associate-/l/ 79.9%

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

   7. metadata-eval 79.9%

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

   8. metadata-eval 79.9%

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

   9. metadata-eval 79.9%

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

   10. /-rgt-identity 79.9%

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

   11. +-commutative 79.9%

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

3. Simplified 79.9%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 4: 78.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

   1. +-commutative 79.8%

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

3. Simplified 79.8%

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

   1. add-log-exp 79.5%

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

6. Applied egg-rr 79.5%

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

   1. rem-log-exp 79.8%

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

   2. cos-sum 99.5%

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

8. Applied egg-rr 80.9%

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

9. Final simplification 80.9%

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

Alternative 5: 78.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

   1. +-commutative 79.8%

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

3. Simplified 79.8%

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

   1. add-log-exp 79.5%

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

6. Applied egg-rr 79.5%

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

   1. add-sqr-sqrt 79.3%

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

   2. sqrt-unprod 79.5%

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

   3. prod-exp 79.5%

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

   4. cos-sum 80.5%

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

   5. cancel-sign-sub-inv 80.5%

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

   6. add-sqr-sqrt 39.5%

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

   7. sqrt-unprod 79.9%

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

   8. sqr-neg 79.9%

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

   9. sqrt-unprod 40.4%

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

   10. add-sqr-sqrt 78.7%

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

   11. cos-diff 79.2%

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

   12. exp-sqrt 79.2%

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

8. Applied egg-rr 80.3%

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

9. Final simplification 80.3%

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

Alternative 6: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{+15} \lor \neg \left(b \leq 2.2 \cdot 10^{-7}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -7.6e+15) (not (<= b 2.2e-7)))
   (* r (/ (sin b) (cos b)))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -7.6e+15) || !(b <= 2.2e-7)) {
		tmp = r * (sin(b) / cos(b));
	} else {
		tmp = b * (r / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-7.6d+15)) .or. (.not. (b <= 2.2d-7))) then
        tmp = r * (sin(b) / cos(b))
    else
        tmp = b * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -7.6e+15) || !(b <= 2.2e-7)) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -7.6e+15) or not (b <= 2.2e-7):
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -7.6e+15) || !(b <= 2.2e-7))
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	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 <= -7.6e+15) || ~((b <= 2.2e-7)))
		tmp = r * (sin(b) / cos(b));
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -7.6e+15], N[Not[LessEqual[b, 2.2e-7]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -7.6e15 or 2.2000000000000001e-7 < b

   1. Initial program 55.0%

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

      1. +-commutative 55.0%

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

   3. Simplified 55.0%

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

   5. Taylor expanded in a around 0 54.7%

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

   if -7.6e15 < b < 2.2000000000000001e-7

   1. Initial program 97.9%

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

      1. associate-*r/ 97.9%

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

      2. /-rgt-identity 97.9%

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

      3. metadata-eval 97.9%

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

      4. associate-/l/ 97.9%

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

      5. associate-*l/ 98.0%

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

      6. associate-/l/ 98.0%

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

      7. metadata-eval 98.0%

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

      8. metadata-eval 98.0%

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

      9. metadata-eval 98.0%

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

      10. /-rgt-identity 98.0%

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

      11. +-commutative 98.0%

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

   3. Simplified 98.0%

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

      1. cos-sum 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in b around 0 97.9%

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

      1. associate-*r/ 97.9%

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

   9. Simplified 97.9%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{+15} \lor \neg \left(b \leq 2.2 \cdot 10^{-7}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{+15} \lor \neg \left(b \leq 2.3 \cdot 10^{-7}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -7.6e+15) (not (<= b 2.3e-7)))
   (* r (/ (sin b) (cos b)))
   (* (sin b) (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -7.6e+15) || !(b <= 2.3e-7)) {
		tmp = r * (sin(b) / cos(b));
	} else {
		tmp = sin(b) * (r / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-7.6d+15)) .or. (.not. (b <= 2.3d-7))) then
        tmp = r * (sin(b) / cos(b))
    else
        tmp = sin(b) * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -7.6e+15) || !(b <= 2.3e-7)) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = Math.sin(b) * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -7.6e+15) or not (b <= 2.3e-7):
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = math.sin(b) * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -7.6e+15) || !(b <= 2.3e-7))
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -7.6e+15) || ~((b <= 2.3e-7)))
		tmp = r * (sin(b) / cos(b));
	else
		tmp = sin(b) * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -7.6e+15], N[Not[LessEqual[b, 2.3e-7]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -7.6e15 or 2.29999999999999995e-7 < b

   1. Initial program 55.0%

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

      1. +-commutative 55.0%

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

   3. Simplified 55.0%

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

   5. Taylor expanded in a around 0 54.7%

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

   if -7.6e15 < b < 2.29999999999999995e-7

   1. Initial program 97.9%

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

      1. associate-*r/ 97.9%

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

      2. /-rgt-identity 97.9%

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

      3. metadata-eval 97.9%

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

      4. associate-/l/ 97.9%

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

      5. associate-*l/ 98.0%

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

      6. associate-/l/ 98.0%

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

      7. metadata-eval 98.0%

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

      8. metadata-eval 98.0%

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

      9. metadata-eval 98.0%

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

      10. /-rgt-identity 98.0%

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

      11. +-commutative 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in b around 0 98.0%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{+15} \lor \neg \left(b \leq 2.3 \cdot 10^{-7}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{+15} \lor \neg \left(b \leq 2.3 \cdot 10^{-7}\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -7.6e+15) (not (<= b 2.3e-7)))
   (* (sin b) (/ r (cos b)))
   (* (sin b) (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -7.6e+15) || !(b <= 2.3e-7)) {
		tmp = sin(b) * (r / cos(b));
	} else {
		tmp = sin(b) * (r / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-7.6d+15)) .or. (.not. (b <= 2.3d-7))) then
        tmp = sin(b) * (r / cos(b))
    else
        tmp = sin(b) * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -7.6e+15) || !(b <= 2.3e-7)) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else {
		tmp = Math.sin(b) * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -7.6e+15) or not (b <= 2.3e-7):
		tmp = math.sin(b) * (r / math.cos(b))
	else:
		tmp = math.sin(b) * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -7.6e+15) || !(b <= 2.3e-7))
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	else
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -7.6e+15) || ~((b <= 2.3e-7)))
		tmp = sin(b) * (r / cos(b));
	else
		tmp = sin(b) * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -7.6e+15], N[Not[LessEqual[b, 2.3e-7]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -7.6e15 or 2.29999999999999995e-7 < b

   1. Initial program 55.0%

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

      1. associate-*r/ 55.0%

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

      2. /-rgt-identity 55.0%

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

      3. metadata-eval 55.0%

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

      4. associate-/l/ 55.0%

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

      5. associate-*l/ 55.1%

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

      6. associate-/l/ 55.1%

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

      7. metadata-eval 55.1%

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

      8. metadata-eval 55.1%

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

      9. metadata-eval 55.1%

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

      10. /-rgt-identity 55.1%

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

      11. +-commutative 55.1%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in a around 0 54.8%

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

   if -7.6e15 < b < 2.29999999999999995e-7

   1. Initial program 97.9%

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

      1. associate-*r/ 97.9%

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

      2. /-rgt-identity 97.9%

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

      3. metadata-eval 97.9%

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

      4. associate-/l/ 97.9%

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

      5. associate-*l/ 98.0%

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

      6. associate-/l/ 98.0%

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

      7. metadata-eval 98.0%

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

      8. metadata-eval 98.0%

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

      9. metadata-eval 98.0%

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

      10. /-rgt-identity 98.0%

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

      11. +-commutative 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in b around 0 98.0%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{+15} \lor \neg \left(b \leq 2.3 \cdot 10^{-7}\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

3. Final simplification 79.8%

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

Alternative 10: 77.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

   1. +-commutative 79.8%

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

3. Simplified 79.8%

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

   1. clear-num 79.7%

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

   2. associate-/r/ 79.8%

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

6. Applied egg-rr 79.8%

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

7. Taylor expanded in r around 0 79.8%

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

9. Simplified 79.9%

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

10. Final simplification 79.9%

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

Alternative 11: 56.0% 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 79.8%

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

   1. +-commutative 79.8%

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

3. Simplified 79.8%

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

5. Taylor expanded in b around 0 62.3%

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

6. Final simplification 62.3%

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

Alternative 12: 56.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+20} \lor \neg \left(b \leq 9000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.6e+20) (not (<= b 9000.0)))
   (* r (sin b))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.6e+20) || !(b <= 9000.0)) {
		tmp = r * sin(b);
	} else {
		tmp = b * (r / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-1.6d+20)) .or. (.not. (b <= 9000.0d0))) then
        tmp = r * sin(b)
    else
        tmp = b * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.6e+20) || !(b <= 9000.0)) {
		tmp = r * Math.sin(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.6e+20) or not (b <= 9000.0):
		tmp = r * math.sin(b)
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.6e+20) || !(b <= 9000.0))
		tmp = Float64(r * sin(b));
	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 <= -1.6e+20) || ~((b <= 9000.0)))
		tmp = r * sin(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.6e+20], N[Not[LessEqual[b, 9000.0]], $MachinePrecision]], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.6e20 or 9e3 < b

   1. Initial program 54.5%

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

      1. +-commutative 54.5%

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

   3. Simplified 54.5%

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

      1. clear-num 54.5%

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

      2. associate-/r/ 54.5%

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

   6. Applied egg-rr 54.5%

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

   7. Taylor expanded in b around 0 11.8%

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

   8. Taylor expanded in a around 0 12.9%

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

   if -1.6e20 < b < 9e3

   1. Initial program 96.8%

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

      1. associate-*r/ 96.8%

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

      2. /-rgt-identity 96.8%

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

      3. metadata-eval 96.8%

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

      4. associate-/l/ 96.8%

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

      5. associate-*l/ 96.9%

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

      6. associate-/l/ 96.9%

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

      7. metadata-eval 96.9%

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

      8. metadata-eval 96.9%

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

      9. metadata-eval 96.9%

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

      10. /-rgt-identity 96.9%

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

      11. +-commutative 96.9%

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

   3. Simplified 96.9%

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

      1. cos-sum 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in b around 0 96.4%

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

      1. associate-*r/ 96.4%

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

   9. Simplified 96.4%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+20} \lor \neg \left(b \leq 9000\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

   1. +-commutative 79.8%

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

3. Simplified 79.8%

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

   1. clear-num 79.7%

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

   2. associate-/r/ 79.8%

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

6. Applied egg-rr 79.8%

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

7. Taylor expanded in b around 0 62.3%

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

8. Taylor expanded in a around 0 42.9%

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

9. Final simplification 42.9%

   \[\leadsto r \cdot \sin b \]
10. Add Preprocessing

Alternative 14: 35.4% 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 79.8%

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

   1. +-commutative 79.8%

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

3. Simplified 79.8%

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

5. Taylor expanded in b around 0 59.2%

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

6. Taylor expanded in a around 0 39.3%

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

7. Final simplification 39.3%

   \[\leadsto r \cdot b \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))