rsin A (should all be same)

Percentage Accurate: 76.1% → 99.5%

Time: 18.6s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. +-commutative 79.6%

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

3. Simplified 79.6%

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

   1. cos-sum 99.5%

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

   2. sub-neg 99.5%

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

5. Applied egg-rr 99.5%

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

   1. +-commutative 99.5%

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

   2. distribute-lft-neg-in 99.5%

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

   3. *-commutative 99.5%

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

   4. fma-def 99.5%

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

   5. *-commutative 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. associate-/l* 79.5%

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

   2. +-commutative 79.5%

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

3. Simplified 79.5%

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

   1. cos-sum 99.5%

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

   2. sub-neg 99.5%

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

5. Applied egg-rr 99.5%

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

   1. +-commutative 99.5%

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

   2. distribute-lft-neg-in 99.5%

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

   3. *-commutative 99.5%

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

   4. fma-def 99.5%

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

   5. *-commutative 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

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 79.6%

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

   1. associate-/l* 79.5%

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

   2. remove-double-neg 79.5%

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

   3. sin-neg 79.5%

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

   4. neg-mul-1 79.5%

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

   5. associate-/r* 79.5%

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

   6. associate-/l* 79.6%

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

   7. *-commutative 79.6%

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

   8. associate-*l/ 79.5%

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

   9. associate-/l* 79.5%

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

   10. sin-neg 79.5%

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

   11. distribute-lft-neg-in 79.5%

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

   12. distribute-rgt-neg-in 79.5%

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

   13. associate-/l* 79.5%

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

   14. metadata-eval 79.5%

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

   15. /-rgt-identity 79.5%

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

   16. +-commutative 79.5%

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

3. Simplified 79.5%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 4: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. associate-/l* 79.5%

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

   2. +-commutative 79.5%

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

3. Simplified 79.5%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

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 79.6%

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

   1. +-commutative 79.6%

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

3. Simplified 79.6%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 6: 76.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

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)) - ((sin(b) * sin(a)) * 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. +-commutative 79.6%

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

3. Simplified 79.6%

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

   1. cos-sum 99.5%

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

   2. sub-neg 99.5%

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

5. Applied egg-rr 99.5%

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

   1. +-commutative 99.5%

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

   2. distribute-lft-neg-in 99.5%

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

   3. *-commutative 99.5%

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

   4. fma-def 99.5%

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

   5. *-commutative 99.5%

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

7. Simplified 99.5%

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

   1. fma-udef 99.5%

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

   2. *-commutative 99.5%

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

   3. distribute-rgt-neg-in 99.5%

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

   4. *-commutative 99.5%

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

   5. +-commutative 99.5%

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

   6. sub-neg 99.5%

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

   7. prod-diff 99.5%

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

   8. *-commutative 99.5%

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

   9. fma-neg 99.5%

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

   10. cos-sum 79.6%

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

   11. *-commutative 79.6%

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

   12. fma-udef 79.6%

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

9. Applied egg-rr 77.8%

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

    1. count-2 77.8%

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

11. Simplified 77.8%

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

    1. add-sqr-sqrt 38.1%

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

    2. sqrt-unprod 79.1%

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

    3. sqr-neg 79.1%

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

    4. sqrt-unprod 41.0%

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

    5. add-sqr-sqrt 80.0%

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

    6. distribute-lft-neg-in 80.0%

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

    7. neg-sub0 80.0%

       \[\leadsto \frac{r \cdot \sin b}{\cos \left(b + a\right) + 2 \cdot \color{blue}{\left(0 - \sin b \cdot \sin a\right)}} \]

13. Applied egg-rr 80.0%

    \[\leadsto \frac{r \cdot \sin b}{\cos \left(b + a\right) + 2 \cdot \color{blue}{\left(0 - \sin b \cdot \sin a\right)}} \]
14. Step-by-step derivation

    1. neg-sub0 80.0%

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

    2. distribute-rgt-neg-in 80.0%

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

15. Simplified 80.0%

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

16. Final simplification 80.0%

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

Alternative 7: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-6} \lor \neg \left(b \leq 16500\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -9e-6) (not (<= b 16500.0)))
   (* (sin b) (/ r (cos b)))
   (/ (* r b) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -9e-6) || !(b <= 16500.0)) {
		tmp = sin(b) * (r / cos(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 <= (-9d-6)) .or. (.not. (b <= 16500.0d0))) then
        tmp = sin(b) * (r / cos(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 <= -9e-6) || !(b <= 16500.0)) {
		tmp = Math.sin(b) * (r / Math.cos(b));
	} else {
		tmp = (r * b) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -9e-6) or not (b <= 16500.0):
		tmp = math.sin(b) * (r / math.cos(b))
	else:
		tmp = (r * b) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -9e-6) || !(b <= 16500.0))
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	else
		tmp = Float64(Float64(r * b) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -9e-6) || ~((b <= 16500.0)))
		tmp = sin(b) * (r / cos(b));
	else
		tmp = (r * b) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -9e-6], N[Not[LessEqual[b, 16500.0]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.00000000000000023e-6 or 16500 < b

   1. Initial program 59.8%

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

      1. associate-/l* 59.8%

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

      2. remove-double-neg 59.8%

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

      3. sin-neg 59.8%

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

      4. neg-mul-1 59.8%

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

      5. associate-/r* 59.8%

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

      6. associate-/l* 59.8%

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

      7. *-commutative 59.8%

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

      8. associate-*l/ 59.8%

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

      9. associate-/l* 59.8%

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

      10. sin-neg 59.8%

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

      11. distribute-lft-neg-in 59.8%

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

      12. distribute-rgt-neg-in 59.8%

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

      13. associate-/l* 59.8%

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

      14. metadata-eval 59.8%

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

      15. /-rgt-identity 59.8%

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

      16. +-commutative 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in a around 0 59.0%

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

      1. associate-/l* 59.0%

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

      2. associate-/r/ 59.0%

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

   6. Simplified 59.0%

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

   if -9.00000000000000023e-6 < b < 16500

   1. Initial program 98.7%

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

      1. associate-/l* 98.6%

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

      2. remove-double-neg 98.6%

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

      3. sin-neg 98.6%

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

      4. neg-mul-1 98.6%

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

      5. associate-/r* 98.6%

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

      6. associate-/l* 98.7%

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

      7. *-commutative 98.7%

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

      8. associate-*l/ 98.6%

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

      9. associate-/l* 98.6%

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

      10. sin-neg 98.6%

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

      11. distribute-lft-neg-in 98.6%

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

      12. distribute-rgt-neg-in 98.6%

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

      13. associate-/l* 98.6%

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

      14. metadata-eval 98.6%

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

      15. /-rgt-identity 98.6%

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

      16. +-commutative 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in b around 0 98.8%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-6} \lor \neg \left(b \leq 16500\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \]

Alternative 8: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 16500:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -6.4e-6)
   (/ r (/ (cos b) (sin b)))
   (if (<= b 16500.0) (/ (* r b) (cos a)) (* (sin b) (/ r (cos b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -6.4e-6) {
		tmp = r / (cos(b) / sin(b));
	} else if (b <= 16500.0) {
		tmp = (r * b) / cos(a);
	} else {
		tmp = sin(b) * (r / cos(b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-6.4d-6)) then
        tmp = r / (cos(b) / sin(b))
    else if (b <= 16500.0d0) then
        tmp = (r * b) / cos(a)
    else
        tmp = sin(b) * (r / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -6.4e-6) {
		tmp = r / (Math.cos(b) / Math.sin(b));
	} else if (b <= 16500.0) {
		tmp = (r * b) / Math.cos(a);
	} else {
		tmp = Math.sin(b) * (r / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -6.4e-6:
		tmp = r / (math.cos(b) / math.sin(b))
	elif b <= 16500.0:
		tmp = (r * b) / math.cos(a)
	else:
		tmp = math.sin(b) * (r / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -6.4e-6)
		tmp = Float64(r / Float64(cos(b) / sin(b)));
	elseif (b <= 16500.0)
		tmp = Float64(Float64(r * b) / cos(a));
	else
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -6.4e-6)
		tmp = r / (cos(b) / sin(b));
	elseif (b <= 16500.0)
		tmp = (r * b) / cos(a);
	else
		tmp = sin(b) * (r / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -6.4e-6], N[(r / N[(N[Cos[b], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 16500.0], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.3999999999999997e-6

   1. Initial program 60.0%

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

      1. associate-/l* 60.1%

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

      2. +-commutative 60.1%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in a around 0 57.9%

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

   if -6.3999999999999997e-6 < b < 16500

   1. Initial program 98.7%

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

      1. associate-/l* 98.6%

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

      2. remove-double-neg 98.6%

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

      3. sin-neg 98.6%

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

      4. neg-mul-1 98.6%

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

      5. associate-/r* 98.6%

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

      6. associate-/l* 98.7%

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

      7. *-commutative 98.7%

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

      8. associate-*l/ 98.6%

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

      9. associate-/l* 98.6%

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

      10. sin-neg 98.6%

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

      11. distribute-lft-neg-in 98.6%

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

      12. distribute-rgt-neg-in 98.6%

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

      13. associate-/l* 98.6%

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

      14. metadata-eval 98.6%

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

      15. /-rgt-identity 98.6%

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

      16. +-commutative 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in b around 0 98.8%

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

   if 16500 < b

   1. Initial program 59.6%

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

      1. associate-/l* 59.5%

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

      2. remove-double-neg 59.5%

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

      3. sin-neg 59.5%

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

      4. neg-mul-1 59.5%

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

      5. associate-/r* 59.5%

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

      6. associate-/l* 59.6%

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

      7. *-commutative 59.6%

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

      8. associate-*l/ 59.5%

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

      9. associate-/l* 59.5%

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

      10. sin-neg 59.5%

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

      11. distribute-lft-neg-in 59.5%

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

      12. distribute-rgt-neg-in 59.5%

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

      13. associate-/l* 59.5%

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

      14. metadata-eval 59.5%

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

      15. /-rgt-identity 59.5%

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

      16. +-commutative 59.5%

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

   3. Simplified 59.5%

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

   4. Taylor expanded in a around 0 60.2%

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

      1. associate-/l* 60.0%

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

      2. associate-/r/ 60.1%

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

   6. Simplified 60.1%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 16500:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]

Alternative 9: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-6}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 16500:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -3e-6)
   (/ r (/ (cos b) (sin b)))
   (if (<= b 16500.0) (/ (* r b) (cos a)) (/ (* r (sin b)) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -3e-6) {
		tmp = r / (cos(b) / sin(b));
	} else if (b <= 16500.0) {
		tmp = (r * b) / cos(a);
	} else {
		tmp = (r * sin(b)) / cos(b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-3d-6)) then
        tmp = r / (cos(b) / sin(b))
    else if (b <= 16500.0d0) then
        tmp = (r * b) / cos(a)
    else
        tmp = (r * sin(b)) / cos(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -3e-6) {
		tmp = r / (Math.cos(b) / Math.sin(b));
	} else if (b <= 16500.0) {
		tmp = (r * b) / Math.cos(a);
	} else {
		tmp = (r * Math.sin(b)) / Math.cos(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -3e-6:
		tmp = r / (math.cos(b) / math.sin(b))
	elif b <= 16500.0:
		tmp = (r * b) / math.cos(a)
	else:
		tmp = (r * math.sin(b)) / math.cos(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -3e-6)
		tmp = Float64(r / Float64(cos(b) / sin(b)));
	elseif (b <= 16500.0)
		tmp = Float64(Float64(r * b) / cos(a));
	else
		tmp = Float64(Float64(r * sin(b)) / cos(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -3e-6)
		tmp = r / (cos(b) / sin(b));
	elseif (b <= 16500.0)
		tmp = (r * b) / cos(a);
	else
		tmp = (r * sin(b)) / cos(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -3e-6], N[(r / N[(N[Cos[b], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 16500.0], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.0000000000000001e-6

   1. Initial program 60.0%

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

      1. associate-/l* 60.1%

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

      2. +-commutative 60.1%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in a around 0 57.9%

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

   if -3.0000000000000001e-6 < b < 16500

   1. Initial program 98.7%

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

      1. associate-/l* 98.6%

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

      2. remove-double-neg 98.6%

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

      3. sin-neg 98.6%

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

      4. neg-mul-1 98.6%

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

      5. associate-/r* 98.6%

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

      6. associate-/l* 98.7%

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

      7. *-commutative 98.7%

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

      8. associate-*l/ 98.6%

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

      9. associate-/l* 98.6%

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

      10. sin-neg 98.6%

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

      11. distribute-lft-neg-in 98.6%

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

      12. distribute-rgt-neg-in 98.6%

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

      13. associate-/l* 98.6%

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

      14. metadata-eval 98.6%

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

      15. /-rgt-identity 98.6%

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

      16. +-commutative 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in b around 0 98.8%

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

   if 16500 < b

   1. Initial program 59.6%

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

      1. +-commutative 59.6%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in a around 0 60.2%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-6}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 16500:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \]

Alternative 10: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{1}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 16500:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -9.2e-5)
   (* r (/ 1.0 (/ (cos b) (sin b))))
   (if (<= b 16500.0) (/ (* r b) (cos a)) (/ (* r (sin b)) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -9.2e-5) {
		tmp = r * (1.0 / (cos(b) / sin(b)));
	} else if (b <= 16500.0) {
		tmp = (r * b) / cos(a);
	} else {
		tmp = (r * sin(b)) / cos(b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-9.2d-5)) then
        tmp = r * (1.0d0 / (cos(b) / sin(b)))
    else if (b <= 16500.0d0) then
        tmp = (r * b) / cos(a)
    else
        tmp = (r * sin(b)) / cos(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -9.2e-5) {
		tmp = r * (1.0 / (Math.cos(b) / Math.sin(b)));
	} else if (b <= 16500.0) {
		tmp = (r * b) / Math.cos(a);
	} else {
		tmp = (r * Math.sin(b)) / Math.cos(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -9.2e-5:
		tmp = r * (1.0 / (math.cos(b) / math.sin(b)))
	elif b <= 16500.0:
		tmp = (r * b) / math.cos(a)
	else:
		tmp = (r * math.sin(b)) / math.cos(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -9.2e-5)
		tmp = Float64(r * Float64(1.0 / Float64(cos(b) / sin(b))));
	elseif (b <= 16500.0)
		tmp = Float64(Float64(r * b) / cos(a));
	else
		tmp = Float64(Float64(r * sin(b)) / cos(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -9.2e-5)
		tmp = r * (1.0 / (cos(b) / sin(b)));
	elseif (b <= 16500.0)
		tmp = (r * b) / cos(a);
	else
		tmp = (r * sin(b)) / cos(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -9.2e-5], N[(r * N[(1.0 / N[(N[Cos[b], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 16500.0], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.20000000000000001e-5

   1. Initial program 60.0%

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

      1. +-commutative 60.0%

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

   3. Simplified 60.0%

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

   4. Taylor expanded in a around 0 57.8%

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

      1. associate-/l* 57.9%

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

      2. div-inv 57.9%

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

   6. Applied egg-rr 57.9%

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

   if -9.20000000000000001e-5 < b < 16500

   1. Initial program 98.7%

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

      1. associate-/l* 98.6%

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

      2. remove-double-neg 98.6%

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

      3. sin-neg 98.6%

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

      4. neg-mul-1 98.6%

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

      5. associate-/r* 98.6%

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

      6. associate-/l* 98.7%

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

      7. *-commutative 98.7%

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

      8. associate-*l/ 98.6%

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

      9. associate-/l* 98.6%

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

      10. sin-neg 98.6%

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

      11. distribute-lft-neg-in 98.6%

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

      12. distribute-rgt-neg-in 98.6%

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

      13. associate-/l* 98.6%

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

      14. metadata-eval 98.6%

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

      15. /-rgt-identity 98.6%

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

      16. +-commutative 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in b around 0 98.8%

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

   if 16500 < b

   1. Initial program 59.6%

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

      1. +-commutative 59.6%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in a around 0 60.2%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{1}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 16500:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \]

Alternative 11: 76.1% 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 79.6%

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

   1. associate-/l* 79.5%

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

   2. remove-double-neg 79.5%

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

   3. sin-neg 79.5%

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

   4. neg-mul-1 79.5%

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

   5. associate-/r* 79.5%

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

   6. associate-/l* 79.6%

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

   7. *-commutative 79.6%

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

   8. associate-*l/ 79.5%

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

   9. associate-/l* 79.5%

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

   10. sin-neg 79.5%

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

   11. distribute-lft-neg-in 79.5%

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

   12. distribute-rgt-neg-in 79.5%

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

   13. associate-/l* 79.5%

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

   14. metadata-eval 79.5%

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

   15. /-rgt-identity 79.5%

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

   16. +-commutative 79.5%

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

3. Simplified 79.5%

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

4. Final simplification 79.5%

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

Alternative 12: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. associate-/l* 79.5%

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

   2. +-commutative 79.5%

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

3. Simplified 79.5%

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

4. Final simplification 79.5%

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

Alternative 13: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

2. Final simplification 79.6%

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

Alternative 14: 74.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.000205 \lor \neg \left(b \leq 16500\right):\\ \;\;\;\;\frac{r}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.000205) (not (<= b 16500.0)))
   (/ r (- (/ 1.0 (tan b)) a))
   (/ (* r b) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.000205) || !(b <= 16500.0)) {
		tmp = r / ((1.0 / tan(b)) - a);
	} else {
		tmp = (r * b) / cos(a);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-0.000205d0)) .or. (.not. (b <= 16500.0d0))) then
        tmp = r / ((1.0d0 / tan(b)) - a)
    else
        tmp = (r * b) / cos(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.000205) || !(b <= 16500.0)) {
		tmp = r / ((1.0 / Math.tan(b)) - a);
	} else {
		tmp = (r * b) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.000205) or not (b <= 16500.0):
		tmp = r / ((1.0 / math.tan(b)) - a)
	else:
		tmp = (r * b) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.000205) || !(b <= 16500.0))
		tmp = Float64(r / Float64(Float64(1.0 / tan(b)) - a));
	else
		tmp = Float64(Float64(r * b) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.000205) || ~((b <= 16500.0)))
		tmp = r / ((1.0 / tan(b)) - a);
	else
		tmp = (r * b) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.000205], N[Not[LessEqual[b, 16500.0]], $MachinePrecision]], N[(r / N[(N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.05e-4 or 16500 < b

   1. Initial program 59.8%

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

      1. associate-/l* 59.8%

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

      2. +-commutative 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in a around 0 55.4%

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

      1. +-commutative 55.4%

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

      2. mul-1-neg 55.4%

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

      3. unsub-neg 55.4%

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

   6. Simplified 55.4%

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

      1. expm1-log1p-u 49.0%

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

      2. expm1-udef 48.9%

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

      3. clear-num 48.8%

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

      4. quot-tan 48.9%

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

   8. Applied egg-rr 48.9%

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

      1. expm1-def 49.0%

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

      2. expm1-log1p 55.4%

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

   10. Simplified 55.4%

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

   if -2.05e-4 < b < 16500

   1. Initial program 98.7%

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

      1. associate-/l* 98.6%

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

      2. remove-double-neg 98.6%

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

      3. sin-neg 98.6%

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

      4. neg-mul-1 98.6%

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

      5. associate-/r* 98.6%

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

      6. associate-/l* 98.7%

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

      7. *-commutative 98.7%

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

      8. associate-*l/ 98.6%

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

      9. associate-/l* 98.6%

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

      10. sin-neg 98.6%

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

      11. distribute-lft-neg-in 98.6%

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

      12. distribute-rgt-neg-in 98.6%

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

      13. associate-/l* 98.6%

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

      14. metadata-eval 98.6%

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

      15. /-rgt-identity 98.6%

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

      16. +-commutative 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in b around 0 98.8%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.000205 \lor \neg \left(b \leq 16500\right):\\ \;\;\;\;\frac{r}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \]

Alternative 15: 74.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\tan b} - a\\ \mathbf{if}\;b \leq -3.7 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{1}{t_0}\\ \mathbf{elif}\;b \leq 16500:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 (tan b)) a)))
   (if (<= b -3.7e-5)
     (* r (/ 1.0 t_0))
     (if (<= b 16500.0) (/ (* r b) (cos a)) (/ r t_0)))))

C

double code(double r, double a, double b) {
	double t_0 = (1.0 / tan(b)) - a;
	double tmp;
	if (b <= -3.7e-5) {
		tmp = r * (1.0 / t_0);
	} else if (b <= 16500.0) {
		tmp = (r * b) / cos(a);
	} else {
		tmp = r / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / tan(b)) - a
    if (b <= (-3.7d-5)) then
        tmp = r * (1.0d0 / t_0)
    else if (b <= 16500.0d0) then
        tmp = (r * b) / cos(a)
    else
        tmp = r / t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (1.0 / Math.tan(b)) - a;
	double tmp;
	if (b <= -3.7e-5) {
		tmp = r * (1.0 / t_0);
	} else if (b <= 16500.0) {
		tmp = (r * b) / Math.cos(a);
	} else {
		tmp = r / t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (1.0 / math.tan(b)) - a
	tmp = 0
	if b <= -3.7e-5:
		tmp = r * (1.0 / t_0)
	elif b <= 16500.0:
		tmp = (r * b) / math.cos(a)
	else:
		tmp = r / t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(1.0 / tan(b)) - a)
	tmp = 0.0
	if (b <= -3.7e-5)
		tmp = Float64(r * Float64(1.0 / t_0));
	elseif (b <= 16500.0)
		tmp = Float64(Float64(r * b) / cos(a));
	else
		tmp = Float64(r / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (1.0 / tan(b)) - a;
	tmp = 0.0;
	if (b <= -3.7e-5)
		tmp = r * (1.0 / t_0);
	elseif (b <= 16500.0)
		tmp = (r * b) / cos(a);
	else
		tmp = r / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]}, If[LessEqual[b, -3.7e-5], N[(r * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 16500.0], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision], N[(r / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.69999999999999981e-5

   1. Initial program 60.0%

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

      1. associate-/l* 60.1%

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

      2. +-commutative 60.1%

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

   3. Simplified 60.1%

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

   4. Taylor expanded in a around 0 55.5%

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

      1. +-commutative 55.5%

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

      2. mul-1-neg 55.5%

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

      3. unsub-neg 55.5%

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

   6. Simplified 55.5%

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

      1. div-inv 55.5%

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

      2. clear-num 55.4%

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

      3. quot-tan 55.4%

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

   8. Applied egg-rr 55.4%

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

   if -3.69999999999999981e-5 < b < 16500

   1. Initial program 98.7%

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

      1. associate-/l* 98.6%

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

      2. remove-double-neg 98.6%

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

      3. sin-neg 98.6%

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

      4. neg-mul-1 98.6%

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

      5. associate-/r* 98.6%

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

      6. associate-/l* 98.7%

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

      7. *-commutative 98.7%

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

      8. associate-*l/ 98.6%

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

      9. associate-/l* 98.6%

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

      10. sin-neg 98.6%

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

      11. distribute-lft-neg-in 98.6%

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

      12. distribute-rgt-neg-in 98.6%

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

      13. associate-/l* 98.6%

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

      14. metadata-eval 98.6%

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

      15. /-rgt-identity 98.6%

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

      16. +-commutative 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in b around 0 98.8%

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

   if 16500 < b

   1. Initial program 59.6%

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

      1. associate-/l* 59.5%

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

      2. +-commutative 59.5%

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

   3. Simplified 59.5%

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

   4. Taylor expanded in a around 0 55.3%

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

      1. +-commutative 55.3%

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

      2. mul-1-neg 55.3%

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

      3. unsub-neg 55.3%

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

   6. Simplified 55.3%

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

      1. expm1-log1p-u 53.3%

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

      2. expm1-udef 53.2%

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

      3. clear-num 53.3%

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

      4. quot-tan 53.3%

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

   8. Applied egg-rr 53.3%

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

      1. expm1-def 53.4%

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

      2. expm1-log1p 55.4%

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

   10. Simplified 55.4%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \mathbf{elif}\;b \leq 16500:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b} - a}\\ \end{array} \]

Alternative 16: 50.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. associate-/l* 79.5%

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

   2. remove-double-neg 79.5%

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

   3. sin-neg 79.5%

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

   4. neg-mul-1 79.5%

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

   5. associate-/r* 79.5%

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

   6. associate-/l* 79.6%

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

   7. *-commutative 79.6%

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

   8. associate-*l/ 79.5%

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

   9. associate-/l* 79.5%

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

   10. sin-neg 79.5%

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

   11. distribute-lft-neg-in 79.5%

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

   12. distribute-rgt-neg-in 79.5%

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

   13. associate-/l* 79.5%

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

   14. metadata-eval 79.5%

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

   15. /-rgt-identity 79.5%

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

   16. +-commutative 79.5%

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

3. Simplified 79.5%

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

4. Taylor expanded in b around 0 52.2%

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

5. Final simplification 52.2%

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

Alternative 17: 50.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.6%

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

   1. associate-/l* 79.5%

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

   2. remove-double-neg 79.5%

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

   3. sin-neg 79.5%

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

   4. neg-mul-1 79.5%

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

   5. associate-/r* 79.5%

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

   6. associate-/l* 79.6%

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

   7. *-commutative 79.6%

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

   8. associate-*l/ 79.5%

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

   9. associate-/l* 79.5%

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

   10. sin-neg 79.5%

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

   11. distribute-lft-neg-in 79.5%

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

   12. distribute-rgt-neg-in 79.5%

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

   13. associate-/l* 79.5%

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

   14. metadata-eval 79.5%

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

   15. /-rgt-identity 79.5%

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

   16. +-commutative 79.5%

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

3. Simplified 79.5%

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

4. Taylor expanded in b around 0 52.2%

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

5. Final simplification 52.2%

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

Alternative 18: 34.2% 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 79.6%

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

   1. associate-/l* 79.5%

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

   2. remove-double-neg 79.5%

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

   3. sin-neg 79.5%

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

   4. neg-mul-1 79.5%

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

   5. associate-/r* 79.5%

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

   6. associate-/l* 79.6%

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

   7. *-commutative 79.6%

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

   8. associate-*l/ 79.5%

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

   9. associate-/l* 79.5%

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

   10. sin-neg 79.5%

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

   11. distribute-lft-neg-in 79.5%

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

   12. distribute-rgt-neg-in 79.5%

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

   13. associate-/l* 79.5%

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

   14. metadata-eval 79.5%

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

   15. /-rgt-identity 79.5%

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

   16. +-commutative 79.5%

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

3. Simplified 79.5%

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

4. Taylor expanded in b around 0 52.2%

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

5. Taylor expanded in a around 0 36.4%

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

   1. *-commutative 36.4%

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

7. Simplified 36.4%

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

8. Final simplification 36.4%

   \[\leadsto r \cdot b \]

Reproduce

herbie shell --seed 2023318 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))