rsin B (should all be same)

Percentage Accurate: 75.8% → 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: 75.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. associate-*r/ 77.8%

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

   2. /-rgt-identity 77.8%

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

   3. metadata-eval 77.8%

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

   4. associate-/l/ 77.8%

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

   5. associate-*l/ 77.8%

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

   6. associate-/l/ 77.8%

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

   7. metadata-eval 77.8%

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

   8. metadata-eval 77.8%

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

   9. metadata-eval 77.8%

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

   10. /-rgt-identity 77.8%

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

   11. +-commutative 77.8%

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

3. Simplified 77.8%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.4%

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

6. Taylor expanded in r around 0 99.4%

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

   1. *-commutative 99.4%

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

   2. *-commutative 99.4%

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

   3. fma-neg 99.4%

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

   4. distribute-rgt-neg-in 99.4%

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

8. Applied egg-rr 99.4%

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

9. Taylor expanded in b around inf 99.4%

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

    1. mul-1-neg 99.4%

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

    2. *-commutative 99.4%

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

    3. distribute-rgt-neg-in 99.4%

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

    4. *-commutative 99.4%

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

    5. fma-def 99.5%

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

    6. *-commutative 99.5%

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

11. Simplified 99.5%

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

12. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. associate-*r/ 77.8%

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

   2. /-rgt-identity 77.8%

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

   3. metadata-eval 77.8%

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

   4. associate-/l/ 77.8%

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

   5. associate-*l/ 77.8%

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

   6. associate-/l/ 77.8%

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

   7. metadata-eval 77.8%

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

   8. metadata-eval 77.8%

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

   9. metadata-eval 77.8%

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

   10. /-rgt-identity 77.8%

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

   11. +-commutative 77.8%

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

3. Simplified 77.8%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.4%

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

6. Taylor expanded in r around 0 99.4%

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

   1. *-commutative 99.4%

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

   2. *-commutative 99.4%

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

   3. fma-neg 99.4%

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

   4. distribute-rgt-neg-in 99.4%

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

8. Applied egg-rr 99.4%

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

9. Final simplification 99.4%

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

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 77.8%

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

   1. +-commutative 77.8%

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

3. Simplified 77.8%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 4: 99.4% accurate, 0.4× speedup

Math

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

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(sin(b) * Float64(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[Sin[b], $MachinePrecision] * N[(r / 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 77.8%

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

   1. associate-*r/ 77.8%

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

   2. /-rgt-identity 77.8%

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

   3. metadata-eval 77.8%

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

   4. associate-/l/ 77.8%

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

   5. associate-*l/ 77.8%

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

   6. associate-/l/ 77.8%

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

   7. metadata-eval 77.8%

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

   8. metadata-eval 77.8%

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

   9. metadata-eval 77.8%

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

   10. /-rgt-identity 77.8%

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

   11. +-commutative 77.8%

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

3. Simplified 77.8%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 5: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. associate-*r/ 77.8%

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

   2. /-rgt-identity 77.8%

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

   3. metadata-eval 77.8%

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

   4. associate-/l/ 77.8%

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

   5. associate-*l/ 77.8%

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

   6. associate-/l/ 77.8%

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

   7. metadata-eval 77.8%

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

   8. metadata-eval 77.8%

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

   9. metadata-eval 77.8%

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

   10. /-rgt-identity 77.8%

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

   11. +-commutative 77.8%

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

3. Simplified 77.8%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.4%

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

6. Taylor expanded in r around 0 99.4%

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

7. Final simplification 99.4%

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

Alternative 6: 76.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, 0\right)} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ (* r (sin b)) (fma (cos b) (cos a) 0.0)))

C

double code(double r, double a, double b) {
	return (r * sin(b)) / fma(cos(b), cos(a), 0.0);
}

Julia

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / fma(cos(b), cos(a), 0.0))
end

Wolfram

code[r_, a_, b_] := N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.8%

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

   1. associate-*r/ 77.8%

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

   2. /-rgt-identity 77.8%

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

   3. metadata-eval 77.8%

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

   4. associate-/l/ 77.8%

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

   5. associate-*l/ 77.8%

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

   6. associate-/l/ 77.8%

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

   7. metadata-eval 77.8%

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

   8. metadata-eval 77.8%

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

   9. metadata-eval 77.8%

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

   10. /-rgt-identity 77.8%

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

   11. +-commutative 77.8%

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

3. Simplified 77.8%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.4%

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

6. Taylor expanded in r around 0 99.4%

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

   1. *-commutative 99.4%

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

   2. *-commutative 99.4%

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

   3. fma-neg 99.4%

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

   4. distribute-rgt-neg-in 99.4%

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

8. Applied egg-rr 99.4%

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

   1. *-commutative 99.4%

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

   2. add-sqr-sqrt 53.1%

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

   3. sqrt-unprod 89.4%

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

   4. sqr-neg 89.4%

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

   5. sqrt-unprod 36.2%

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

   6. add-sqr-sqrt 77.5%

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

   7. sin-mult 78.5%

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

   8. cos-sum 77.4%

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

   9. div-sub 77.4%

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

10. Applied egg-rr 78.6%

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

    1. +-inverses 78.6%

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

12. Simplified 78.6%

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

13. Final simplification 78.6%

    \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, 0\right)} \]

Alternative 7: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 22:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -2.9e-5)
   (* r (/ (sin b) (cos a)))
   (if (<= a 22.0) (* r (tan b)) (* (sin b) (/ r (cos a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -2.9e-5) {
		tmp = r * (sin(b) / cos(a));
	} else if (a <= 22.0) {
		tmp = r * tan(b);
	} else {
		tmp = sin(b) * (r / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-2.9d-5)) then
        tmp = r * (sin(b) / cos(a))
    else if (a <= 22.0d0) then
        tmp = r * tan(b)
    else
        tmp = sin(b) * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -2.9e-5) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else if (a <= 22.0) {
		tmp = r * Math.tan(b);
	} else {
		tmp = Math.sin(b) * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -2.9e-5:
		tmp = r * (math.sin(b) / math.cos(a))
	elif a <= 22.0:
		tmp = r * math.tan(b)
	else:
		tmp = math.sin(b) * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -2.9e-5)
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	elseif (a <= 22.0)
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(sin(b) * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -2.9e-5)
		tmp = r * (sin(b) / cos(a));
	elseif (a <= 22.0)
		tmp = r * tan(b);
	else
		tmp = sin(b) * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -2.9e-5], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 22.0], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.9e-5

   1. Initial program 61.7%

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

      1. +-commutative 61.7%

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

   3. Simplified 61.7%

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

   4. Taylor expanded in b around 0 60.1%

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

   if -2.9e-5 < a < 22

   1. Initial program 98.3%

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

      1. +-commutative 98.3%

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

   3. Simplified 98.3%

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

      1. add-log-exp 72.0%

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

   5. Applied egg-rr 72.0%

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

   6. Taylor expanded in a around 0 72.0%

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

      1. rem-log-exp 72.0%

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

      2. *-un-lft-identity 72.0%

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

      3. log-prod 72.0%

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

      4. metadata-eval 72.0%

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

      5. add-log-exp 72.0%

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

      6. rem-log-exp 98.3%

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

      7. quot-tan 98.4%

         \[\leadsto r \cdot \left(0 + \color{blue}{\tan b}\right) \]

   8. Applied egg-rr 98.4%

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

      1. +-lft-identity 98.4%

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

   10. Simplified 98.4%

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

   if 22 < a

   1. Initial program 52.1%

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

      1. associate-*r/ 52.1%

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

      2. /-rgt-identity 52.1%

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

      3. metadata-eval 52.1%

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

      4. associate-/l/ 52.1%

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

      5. associate-*l/ 52.2%

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

      6. associate-/l/ 52.2%

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

      7. metadata-eval 52.2%

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

      8. metadata-eval 52.2%

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

      9. metadata-eval 52.2%

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

      10. /-rgt-identity 52.2%

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

      11. +-commutative 52.2%

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

   3. Simplified 52.2%

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

   4. Taylor expanded in b around 0 52.9%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 22:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 8: 75.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

2. Final simplification 77.8%

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

Alternative 9: 75.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. associate-*r/ 77.8%

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

   2. /-rgt-identity 77.8%

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

   3. metadata-eval 77.8%

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

   4. associate-/l/ 77.8%

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

   5. associate-*l/ 77.8%

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

   6. associate-/l/ 77.8%

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

   7. metadata-eval 77.8%

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

   8. metadata-eval 77.8%

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

   9. metadata-eval 77.8%

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

   10. /-rgt-identity 77.8%

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

   11. +-commutative 77.8%

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

3. Simplified 77.8%

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

4. Final simplification 77.8%

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

Alternative 10: 75.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.000245 \lor \neg \left(b \leq 0.0015\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left(b \cdot \frac{1}{\cos a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.000245) (not (<= b 0.0015)))
   (* r (tan b))
   (* r (* b (/ 1.0 (cos a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.000245) || !(b <= 0.0015)) {
		tmp = r * tan(b);
	} else {
		tmp = r * (b * (1.0 / 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 <= (-0.000245d0)) .or. (.not. (b <= 0.0015d0))) then
        tmp = r * tan(b)
    else
        tmp = r * (b * (1.0d0 / cos(a)))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.000245) || !(b <= 0.0015)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = r * (b * (1.0 / Math.cos(a)));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.000245) or not (b <= 0.0015):
		tmp = r * math.tan(b)
	else:
		tmp = r * (b * (1.0 / math.cos(a)))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.000245) || !(b <= 0.0015))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(r * Float64(b * Float64(1.0 / cos(a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.000245) || ~((b <= 0.0015)))
		tmp = r * tan(b);
	else
		tmp = r * (b * (1.0 / cos(a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.000245], N[Not[LessEqual[b, 0.0015]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(b * N[(1.0 / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.4499999999999999e-4 or 0.0015 < b

   1. Initial program 59.1%

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

      1. +-commutative 59.1%

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

   3. Simplified 59.1%

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

      1. add-log-exp 58.8%

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

   5. Applied egg-rr 58.8%

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

   6. Taylor expanded in a around 0 58.4%

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

      1. rem-log-exp 58.4%

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

      2. *-un-lft-identity 58.4%

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

      3. log-prod 58.4%

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

      4. metadata-eval 58.4%

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

      5. add-log-exp 58.4%

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

      6. rem-log-exp 58.8%

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

      7. quot-tan 58.9%

         \[\leadsto r \cdot \left(0 + \color{blue}{\tan b}\right) \]

   8. Applied egg-rr 58.9%

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

      1. +-lft-identity 58.9%

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

   10. Simplified 58.9%

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

   if -2.4499999999999999e-4 < b < 0.0015

   1. Initial program 98.2%

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

      1. +-commutative 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in b around 0 98.2%

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

      1. clear-num 98.1%

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

      2. associate-/r/ 98.3%

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

   6. Applied egg-rr 98.3%

      \[\leadsto r \cdot \color{blue}{\left(\frac{1}{\cos a} \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.000245 \lor \neg \left(b \leq 0.0015\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \left(b \cdot \frac{1}{\cos a}\right)\\ \end{array} \]

Alternative 11: 75.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{-5} \lor \neg \left(b \leq 0.00155\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -8.6e-5) (not (<= b 0.00155)))
   (* r (tan b))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -8.6e-5) || !(b <= 0.00155)) {
		tmp = r * tan(b);
	} else {
		tmp = r * (b / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-8.6d-5)) .or. (.not. (b <= 0.00155d0))) then
        tmp = r * tan(b)
    else
        tmp = r * (b / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -8.6e-5) || !(b <= 0.00155)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -8.6e-5) or not (b <= 0.00155):
		tmp = r * math.tan(b)
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -8.6e-5) || !(b <= 0.00155))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(r * Float64(b / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -8.6e-5) || ~((b <= 0.00155)))
		tmp = r * tan(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -8.6e-5], N[Not[LessEqual[b, 0.00155]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -8.6000000000000003e-5 or 0.00154999999999999995 < b

   1. Initial program 59.1%

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

      1. +-commutative 59.1%

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

   3. Simplified 59.1%

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

      1. add-log-exp 58.8%

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

   5. Applied egg-rr 58.8%

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

   6. Taylor expanded in a around 0 58.4%

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

      1. rem-log-exp 58.4%

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

      2. *-un-lft-identity 58.4%

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

      3. log-prod 58.4%

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

      4. metadata-eval 58.4%

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

      5. add-log-exp 58.4%

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

      6. rem-log-exp 58.8%

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

      7. quot-tan 58.9%

         \[\leadsto r \cdot \left(0 + \color{blue}{\tan b}\right) \]

   8. Applied egg-rr 58.9%

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

      1. +-lft-identity 58.9%

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

   10. Simplified 58.9%

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

   if -8.6000000000000003e-5 < b < 0.00154999999999999995

   1. Initial program 98.2%

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

      1. +-commutative 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in b around 0 98.2%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.6 \cdot 10^{-5} \lor \neg \left(b \leq 0.00155\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 12: 38.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. associate-*r/ 77.8%

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

   2. /-rgt-identity 77.8%

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

   3. metadata-eval 77.8%

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

   4. associate-/l/ 77.8%

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

   5. associate-*l/ 77.8%

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

   6. associate-/l/ 77.8%

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

   7. metadata-eval 77.8%

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

   8. metadata-eval 77.8%

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

   9. metadata-eval 77.8%

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

   10. /-rgt-identity 77.8%

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

   11. +-commutative 77.8%

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

3. Simplified 77.8%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.4%

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

6. Taylor expanded in a around 0 62.3%

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

7. Taylor expanded in b around 0 37.3%

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

8. Final simplification 37.3%

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

Alternative 13: 59.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. +-commutative 77.8%

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

3. Simplified 77.8%

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

   1. add-log-exp 46.8%

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

5. Applied egg-rr 46.8%

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

6. Taylor expanded in a around 0 46.1%

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

   1. rem-log-exp 46.1%

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

   2. *-un-lft-identity 46.1%

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

   3. log-prod 46.1%

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

   4. metadata-eval 46.1%

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

   5. add-log-exp 46.1%

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

   6. rem-log-exp 61.3%

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

   7. quot-tan 61.3%

      \[\leadsto r \cdot \left(0 + \color{blue}{\tan b}\right) \]

8. Applied egg-rr 61.3%

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

   1. +-lft-identity 61.3%

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

10. Simplified 61.3%

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

11. Final simplification 61.3%

    \[\leadsto r \cdot \tan b \]

Alternative 14: 34.9% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.8%

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

   1. +-commutative 77.8%

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

3. Simplified 77.8%

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

4. Taylor expanded in b around 0 48.9%

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

5. Taylor expanded in a around 0 32.5%

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

6. Final simplification 32.5%

   \[\leadsto r \cdot b \]

Reproduce

herbie shell --seed 2023322 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))