rsin A (should all be same)

Percentage Accurate: 77.3% → 99.5%

Time: 16.8s

Alternatives: 14

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: 77.3% 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, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

   1. associate-/l* 79.7%

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

   2. remove-double-neg 79.7%

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

   3. sin-neg 79.7%

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

   4. neg-mul-1 79.7%

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

   5. associate-/r* 79.7%

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

   6. associate-/l* 79.8%

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

   7. *-commutative 79.8%

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

   8. associate-*l/ 79.8%

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

   9. associate-/l* 79.8%

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

   10. sin-neg 79.8%

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

   11. distribute-lft-neg-in 79.8%

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

   12. distribute-rgt-neg-in 79.8%

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

   13. associate-/l* 79.8%

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

   14. metadata-eval 79.8%

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

   15. /-rgt-identity 79.8%

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

   16. +-commutative 79.8%

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

3. Simplified 79.8%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

   1. prod-diff 99.5%

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

   2. *-commutative 99.5%

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

   3. fma-def 99.5%

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

   4. associate-+l+ 99.5%

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

   5. *-commutative 99.5%

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

   6. distribute-rgt-neg-in 99.5%

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

   7. *-commutative 99.5%

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

8. Applied egg-rr 99.5%

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

   1. fma-def 99.5%

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

   2. fma-def 99.5%

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

   3. *-commutative 99.5%

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

10. Simplified 99.5%

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

11. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

   1. associate-/l* 79.7%

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

   2. remove-double-neg 79.7%

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

   3. sin-neg 79.7%

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

   4. neg-mul-1 79.7%

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

   5. associate-/r* 79.7%

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

   6. associate-/l* 79.8%

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

   7. *-commutative 79.8%

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

   8. associate-*l/ 79.8%

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

   9. associate-/l* 79.8%

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

   10. sin-neg 79.8%

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

   11. distribute-lft-neg-in 79.8%

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

   12. distribute-rgt-neg-in 79.8%

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

   13. associate-/l* 79.8%

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

   14. metadata-eval 79.8%

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

   15. /-rgt-identity 79.8%

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

   16. +-commutative 79.8%

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

3. Simplified 79.8%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 3: 78.1% accurate, 0.5× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

   1. associate-/l* 79.7%

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

   2. remove-double-neg 79.7%

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

   3. sin-neg 79.7%

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

   4. neg-mul-1 79.7%

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

   5. associate-/r* 79.7%

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

   6. associate-/l* 79.8%

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

   7. *-commutative 79.8%

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

   8. associate-*l/ 79.8%

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

   9. associate-/l* 79.8%

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

   10. sin-neg 79.8%

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

   11. distribute-lft-neg-in 79.8%

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

   12. distribute-rgt-neg-in 79.8%

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

   13. associate-/l* 79.8%

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

   14. metadata-eval 79.8%

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

   15. /-rgt-identity 79.8%

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

   16. +-commutative 79.8%

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

3. Simplified 79.8%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

   1. fma-neg 99.5%

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

   2. *-commutative 99.5%

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

   3. distribute-rgt-neg-in 99.5%

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

8. Applied egg-rr 99.5%

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

   1. *-commutative 99.5%

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

   2. add-sqr-sqrt 51.6%

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

   3. sqrt-unprod 92.4%

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

   4. sqr-neg 92.4%

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

   5. sqrt-unprod 40.8%

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

   6. add-sqr-sqrt 79.5%

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

   7. sin-mult 80.3%

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

   8. cos-diff 79.7%

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

   9. add-sqr-sqrt 40.9%

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

   10. sqrt-unprod 80.3%

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

   11. sqr-neg 80.3%

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

   12. sqrt-unprod 39.4%

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

   13. add-sqr-sqrt 80.7%

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

   14. cancel-sign-sub-inv 80.7%

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

   15. cos-sum 80.3%

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

10. Applied egg-rr 80.3%

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

    1. +-inverses 80.3%

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

    2. metadata-eval 80.3%

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

12. Simplified 80.3%

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

13. Final simplification 80.3%

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

Alternative 4: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{-5} \lor \neg \left(a \leq 0.00075\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -8e-5) (not (<= a 0.00075)))
   (* r (/ (sin b) (cos a)))
   (* r (/ 1.0 (- (/ 1.0 (tan b)) a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -8e-5) || !(a <= 0.00075)) {
		tmp = r * (sin(b) / cos(a));
	} else {
		tmp = r * (1.0 / ((1.0 / tan(b)) - a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-8d-5)) .or. (.not. (a <= 0.00075d0))) then
        tmp = r * (sin(b) / cos(a))
    else
        tmp = r * (1.0d0 / ((1.0d0 / tan(b)) - a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -8e-5) || !(a <= 0.00075)) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else {
		tmp = r * (1.0 / ((1.0 / Math.tan(b)) - a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -8e-5) or not (a <= 0.00075):
		tmp = r * (math.sin(b) / math.cos(a))
	else:
		tmp = r * (1.0 / ((1.0 / math.tan(b)) - a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -8e-5) || !(a <= 0.00075))
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	else
		tmp = Float64(r * Float64(1.0 / Float64(Float64(1.0 / tan(b)) - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -8e-5) || ~((a <= 0.00075)))
		tmp = r * (sin(b) / cos(a));
	else
		tmp = r * (1.0 / ((1.0 / tan(b)) - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -8e-5], N[Not[LessEqual[a, 0.00075]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(1.0 / N[(N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8.00000000000000065e-5 or 7.5000000000000002e-4 < a

   1. Initial program 63.3%

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

      1. associate-/l* 63.3%

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

      2. remove-double-neg 63.3%

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

      3. sin-neg 63.3%

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

      4. neg-mul-1 63.3%

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

      5. associate-/r* 63.3%

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

      6. associate-/l* 63.3%

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

      7. *-commutative 63.3%

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

      8. associate-*l/ 63.3%

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

      9. associate-/l* 63.3%

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

      10. sin-neg 63.3%

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

      11. distribute-lft-neg-in 63.3%

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

      12. distribute-rgt-neg-in 63.3%

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

      13. associate-/l* 63.3%

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

      14. metadata-eval 63.3%

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

      15. /-rgt-identity 63.3%

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

      16. +-commutative 63.3%

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

   3. Simplified 63.3%

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

   5. Taylor expanded in b around 0 62.6%

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

   if -8.00000000000000065e-5 < a < 7.5000000000000002e-4

   1. Initial program 98.9%

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

      1. associate-/l* 98.7%

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

      2. +-commutative 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in a around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. mul-1-neg 99.3%

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

      3. unsub-neg 99.3%

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

   7. Simplified 99.3%

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

      1. div-inv 99.3%

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

      2. clear-num 99.3%

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

      3. quot-tan 99.4%

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

   9. Applied egg-rr 99.4%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{-5} \lor \neg \left(a \leq 0.00075\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.000185:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{elif}\;a \leq 0.0006:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.000185)
   (/ r (/ (cos a) (sin b)))
   (if (<= a 0.0006)
     (* r (/ 1.0 (- (/ 1.0 (tan b)) a)))
     (* r (/ (sin b) (cos a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.000185) {
		tmp = r / (cos(a) / sin(b));
	} else if (a <= 0.0006) {
		tmp = r * (1.0 / ((1.0 / tan(b)) - a));
	} else {
		tmp = r * (sin(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 (a <= (-0.000185d0)) then
        tmp = r / (cos(a) / sin(b))
    else if (a <= 0.0006d0) then
        tmp = r * (1.0d0 / ((1.0d0 / tan(b)) - a))
    else
        tmp = r * (sin(b) / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.000185) {
		tmp = r / (Math.cos(a) / Math.sin(b));
	} else if (a <= 0.0006) {
		tmp = r * (1.0 / ((1.0 / Math.tan(b)) - a));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -0.000185:
		tmp = r / (math.cos(a) / math.sin(b))
	elif a <= 0.0006:
		tmp = r * (1.0 / ((1.0 / math.tan(b)) - a))
	else:
		tmp = r * (math.sin(b) / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -0.000185)
		tmp = Float64(r / Float64(cos(a) / sin(b)));
	elseif (a <= 0.0006)
		tmp = Float64(r * Float64(1.0 / Float64(Float64(1.0 / tan(b)) - a)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -0.000185)
		tmp = r / (cos(a) / sin(b));
	elseif (a <= 0.0006)
		tmp = r * (1.0 / ((1.0 / tan(b)) - a));
	else
		tmp = r * (sin(b) / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -0.000185], N[(r / N[(N[Cos[a], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.0006], N[(r * N[(1.0 / N[(N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.85e-4

   1. Initial program 68.3%

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

      1. associate-/l* 68.3%

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

      2. +-commutative 68.3%

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

   3. Simplified 68.3%

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

   5. Taylor expanded in b around 0 67.4%

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

   if -1.85e-4 < a < 5.99999999999999947e-4

   1. Initial program 98.9%

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

      1. associate-/l* 98.7%

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

      2. +-commutative 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in a around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. mul-1-neg 99.3%

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

      3. unsub-neg 99.3%

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

   7. Simplified 99.3%

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

      1. div-inv 99.3%

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

      2. clear-num 99.3%

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

      3. quot-tan 99.4%

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

   9. Applied egg-rr 99.4%

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

   if 5.99999999999999947e-4 < a

   1. Initial program 59.2%

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

      1. associate-/l* 59.2%

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

      2. remove-double-neg 59.2%

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

      3. sin-neg 59.2%

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

      4. neg-mul-1 59.2%

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

      5. associate-/r* 59.2%

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

      6. associate-/l* 59.2%

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

      7. *-commutative 59.2%

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

      8. associate-*l/ 59.2%

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

      9. associate-/l* 59.2%

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

      10. sin-neg 59.2%

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

      11. distribute-lft-neg-in 59.2%

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

      12. distribute-rgt-neg-in 59.2%

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

      13. associate-/l* 59.2%

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

      14. metadata-eval 59.2%

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

      15. /-rgt-identity 59.2%

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

      16. +-commutative 59.2%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in b around 0 58.8%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.000185:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{elif}\;a \leq 0.0006:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -7.6e+15)
   (/ r (/ (cos b) (sin b)))
   (if (<= b 2.3e-7) (* b (/ r (cos a))) (* r (/ (sin b) (cos b))))))

C

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

Python

def code(r, a, b):
	tmp = 0
	if b <= -7.6e+15:
		tmp = r / (math.cos(b) / math.sin(b))
	elif b <= 2.3e-7:
		tmp = b * (r / 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 <= -7.6e+15)
		tmp = Float64(r / Float64(cos(b) / sin(b)));
	elseif (b <= 2.3e-7)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	end
	return tmp
end

MATLAB

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

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -7.6e+15], N[(r / N[(N[Cos[b], $MachinePrecision] / N[Sin[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.3e-7], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.6e15

   1. Initial program 49.0%

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

      1. associate-/l* 49.0%

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

      2. +-commutative 49.0%

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

   3. Simplified 49.0%

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

   5. Taylor expanded in a around 0 48.5%

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

   if -7.6e15 < b < 2.29999999999999995e-7

   1. Initial program 97.9%

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

      1. associate-/l* 97.8%

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

      2. remove-double-neg 97.8%

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

      3. sin-neg 97.8%

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

      4. neg-mul-1 97.8%

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

      5. associate-/r* 97.8%

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

      6. associate-/l* 97.9%

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

      7. *-commutative 97.9%

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

      8. associate-*l/ 97.9%

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

      9. associate-/l* 97.9%

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

      10. sin-neg 97.9%

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

      11. distribute-lft-neg-in 97.9%

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

      12. distribute-rgt-neg-in 97.9%

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

      13. associate-/l* 97.9%

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

      14. metadata-eval 97.9%

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

      15. /-rgt-identity 97.9%

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

      16. +-commutative 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in b around 0 97.9%

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

      1. *-commutative 97.9%

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

      2. clear-num 97.8%

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

      3. un-div-inv 97.7%

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

   7. Applied egg-rr 97.7%

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

      1. associate-/r/ 97.9%

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

   9. Simplified 97.9%

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

   if 2.29999999999999995e-7 < b

   1. Initial program 59.9%

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

      1. associate-/l* 60.0%

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

      2. remove-double-neg 60.0%

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

      3. sin-neg 60.0%

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

      4. neg-mul-1 60.0%

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

      5. associate-/r* 60.0%

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

      6. associate-/l* 59.9%

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

      7. *-commutative 59.9%

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

      8. associate-*l/ 60.0%

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

      9. associate-/l* 60.0%

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

      10. sin-neg 60.0%

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

      11. distribute-lft-neg-in 60.0%

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

      12. distribute-rgt-neg-in 60.0%

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

      13. associate-/l* 60.0%

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

      14. metadata-eval 60.0%

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

      15. /-rgt-identity 60.0%

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

      16. +-commutative 60.0%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in a around 0 59.9%

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

4. Final simplification 79.7%

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

Alternative 7: 77.3% 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 79.8%

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

   1. associate-/l* 79.7%

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

   2. +-commutative 79.7%

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

3. Simplified 79.7%

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

   1. associate-/r/ 79.9%

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

6. Applied egg-rr 79.9%

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

7. Final simplification 79.9%

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

Alternative 8: 74.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\tan b} - a\\ \mathbf{if}\;b \leq -1.6 \cdot 10^{+20}:\\ \;\;\;\;\frac{r}{t_0}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-7}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 (tan b)) a)))
   (if (<= b -1.6e+20)
     (/ r t_0)
     (if (<= b 2.3e-7) (* b (/ r (cos a))) (* r (/ 1.0 t_0))))))

C

double code(double r, double a, double b) {
	double t_0 = (1.0 / tan(b)) - a;
	double tmp;
	if (b <= -1.6e+20) {
		tmp = r / t_0;
	} else if (b <= 2.3e-7) {
		tmp = b * (r / cos(a));
	} else {
		tmp = r * (1.0 / t_0);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / tan(b)) - a
    if (b <= (-1.6d+20)) then
        tmp = r / t_0
    else if (b <= 2.3d-7) then
        tmp = b * (r / cos(a))
    else
        tmp = r * (1.0d0 / t_0)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (1.0 / Math.tan(b)) - a;
	double tmp;
	if (b <= -1.6e+20) {
		tmp = r / t_0;
	} else if (b <= 2.3e-7) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = r * (1.0 / t_0);
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (1.0 / math.tan(b)) - a
	tmp = 0
	if b <= -1.6e+20:
		tmp = r / t_0
	elif b <= 2.3e-7:
		tmp = b * (r / math.cos(a))
	else:
		tmp = r * (1.0 / t_0)
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(1.0 / tan(b)) - a)
	tmp = 0.0
	if (b <= -1.6e+20)
		tmp = Float64(r / t_0);
	elseif (b <= 2.3e-7)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = Float64(r * Float64(1.0 / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (1.0 / tan(b)) - a;
	tmp = 0.0;
	if (b <= -1.6e+20)
		tmp = r / t_0;
	elseif (b <= 2.3e-7)
		tmp = b * (r / cos(a));
	else
		tmp = r * (1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]}, If[LessEqual[b, -1.6e+20], N[(r / t$95$0), $MachinePrecision], If[LessEqual[b, 2.3e-7], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.6e20

   1. Initial program 50.7%

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

      1. associate-/l* 50.7%

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

      2. +-commutative 50.7%

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

   3. Simplified 50.7%

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

   5. Taylor expanded in a around 0 48.8%

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

      1. +-commutative 48.8%

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

      2. mul-1-neg 48.8%

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

      3. unsub-neg 48.8%

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

   7. Simplified 48.8%

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

      1. expm1-log1p-u 28.6%

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

      2. expm1-udef 15.1%

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

      3. clear-num 15.1%

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

      4. quot-tan 15.1%

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

   9. Applied egg-rr 15.1%

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

       1. expm1-def 28.6%

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

       2. expm1-log1p 48.8%

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

   11. Simplified 48.8%

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

   if -1.6e20 < b < 2.29999999999999995e-7

   1. Initial program 96.7%

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

      1. associate-/l* 96.6%

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

      2. remove-double-neg 96.6%

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

      3. sin-neg 96.6%

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

      4. neg-mul-1 96.6%

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

      5. associate-/r* 96.6%

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

      6. associate-/l* 96.7%

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

      7. *-commutative 96.7%

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

      8. associate-*l/ 96.8%

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

      9. associate-/l* 96.8%

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

      10. sin-neg 96.8%

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

      11. distribute-lft-neg-in 96.8%

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

      12. distribute-rgt-neg-in 96.8%

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

      13. associate-/l* 96.8%

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

      14. metadata-eval 96.8%

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

      15. /-rgt-identity 96.8%

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

      16. +-commutative 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in b around 0 96.7%

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

      1. *-commutative 96.7%

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

      2. clear-num 96.6%

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

      3. un-div-inv 96.6%

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

   7. Applied egg-rr 96.6%

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

      1. associate-/r/ 96.8%

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

   9. Simplified 96.8%

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

   if 2.29999999999999995e-7 < b

   1. Initial program 59.9%

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

      1. associate-/l* 60.0%

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

      2. +-commutative 60.0%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in a around 0 55.8%

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

      1. +-commutative 55.8%

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

      2. mul-1-neg 55.8%

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

      3. unsub-neg 55.8%

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

   7. Simplified 55.8%

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

      1. div-inv 55.7%

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

      2. clear-num 55.8%

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

      3. quot-tan 55.9%

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

   9. Applied egg-rr 55.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+20}:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b} - a}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-7}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\tan b} - a\\ \mathbf{if}\;b \leq -1.6 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{\frac{t_0}{r}}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-7}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 (tan b)) a)))
   (if (<= b -1.6e+20)
     (/ 1.0 (/ t_0 r))
     (if (<= b 2.3e-7) (* b (/ r (cos a))) (* r (/ 1.0 t_0))))))

C

double code(double r, double a, double b) {
	double t_0 = (1.0 / tan(b)) - a;
	double tmp;
	if (b <= -1.6e+20) {
		tmp = 1.0 / (t_0 / r);
	} else if (b <= 2.3e-7) {
		tmp = b * (r / cos(a));
	} else {
		tmp = r * (1.0 / t_0);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / tan(b)) - a
    if (b <= (-1.6d+20)) then
        tmp = 1.0d0 / (t_0 / r)
    else if (b <= 2.3d-7) then
        tmp = b * (r / cos(a))
    else
        tmp = r * (1.0d0 / t_0)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (1.0 / Math.tan(b)) - a;
	double tmp;
	if (b <= -1.6e+20) {
		tmp = 1.0 / (t_0 / r);
	} else if (b <= 2.3e-7) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = r * (1.0 / t_0);
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (1.0 / math.tan(b)) - a
	tmp = 0
	if b <= -1.6e+20:
		tmp = 1.0 / (t_0 / r)
	elif b <= 2.3e-7:
		tmp = b * (r / math.cos(a))
	else:
		tmp = r * (1.0 / t_0)
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(1.0 / tan(b)) - a)
	tmp = 0.0
	if (b <= -1.6e+20)
		tmp = Float64(1.0 / Float64(t_0 / r));
	elseif (b <= 2.3e-7)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = Float64(r * Float64(1.0 / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (1.0 / tan(b)) - a;
	tmp = 0.0;
	if (b <= -1.6e+20)
		tmp = 1.0 / (t_0 / r);
	elseif (b <= 2.3e-7)
		tmp = b * (r / cos(a));
	else
		tmp = r * (1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]}, If[LessEqual[b, -1.6e+20], N[(1.0 / N[(t$95$0 / r), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.3e-7], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.6e20

   1. Initial program 50.7%

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

      1. associate-/l* 50.7%

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

      2. +-commutative 50.7%

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

   3. Simplified 50.7%

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

   5. Taylor expanded in a around 0 48.8%

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

      1. +-commutative 48.8%

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

      2. mul-1-neg 48.8%

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

      3. unsub-neg 48.8%

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

   7. Simplified 48.8%

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

      1. clear-num 48.8%

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

      2. inv-pow 48.8%

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

      3. clear-num 48.7%

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

      4. quot-tan 48.9%

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

   9. Applied egg-rr 48.9%

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

       1. unpow-1 48.9%

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

   11. Simplified 48.9%

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

   if -1.6e20 < b < 2.29999999999999995e-7

   1. Initial program 96.7%

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

      1. associate-/l* 96.6%

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

      2. remove-double-neg 96.6%

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

      3. sin-neg 96.6%

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

      4. neg-mul-1 96.6%

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

      5. associate-/r* 96.6%

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

      6. associate-/l* 96.7%

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

      7. *-commutative 96.7%

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

      8. associate-*l/ 96.8%

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

      9. associate-/l* 96.8%

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

      10. sin-neg 96.8%

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

      11. distribute-lft-neg-in 96.8%

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

      12. distribute-rgt-neg-in 96.8%

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

      13. associate-/l* 96.8%

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

      14. metadata-eval 96.8%

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

      15. /-rgt-identity 96.8%

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

      16. +-commutative 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in b around 0 96.7%

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

      1. *-commutative 96.7%

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

      2. clear-num 96.6%

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

      3. un-div-inv 96.6%

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

   7. Applied egg-rr 96.6%

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

      1. associate-/r/ 96.8%

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

   9. Simplified 96.8%

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

   if 2.29999999999999995e-7 < b

   1. Initial program 59.9%

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

      1. associate-/l* 60.0%

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

      2. +-commutative 60.0%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in a around 0 55.8%

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

      1. +-commutative 55.8%

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

      2. mul-1-neg 55.8%

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

      3. unsub-neg 55.8%

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

   7. Simplified 55.8%

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

      1. div-inv 55.7%

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

      2. clear-num 55.8%

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

      3. quot-tan 55.9%

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

   9. Applied egg-rr 55.9%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{\tan b} - a}{r}}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-7}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+20} \lor \neg \left(b \leq 2.3 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{r}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.6e+20) (not (<= b 2.3e-7)))
   (/ r (- (/ 1.0 (tan b)) a))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.6e+20) || !(b <= 2.3e-7)) {
		tmp = r / ((1.0 / tan(b)) - a);
	} else {
		tmp = b * (r / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-1.6d+20)) .or. (.not. (b <= 2.3d-7))) then
        tmp = r / ((1.0d0 / tan(b)) - a)
    else
        tmp = b * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.6e+20) || !(b <= 2.3e-7)) {
		tmp = r / ((1.0 / Math.tan(b)) - a);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.6e+20) or not (b <= 2.3e-7):
		tmp = r / ((1.0 / math.tan(b)) - a)
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.6e+20) || !(b <= 2.3e-7))
		tmp = Float64(r / Float64(Float64(1.0 / tan(b)) - a));
	else
		tmp = Float64(b * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -1.6e+20) || ~((b <= 2.3e-7)))
		tmp = r / ((1.0 / tan(b)) - a);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.6e+20], N[Not[LessEqual[b, 2.3e-7]], $MachinePrecision]], N[(r / N[(N[(1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.6e20 or 2.29999999999999995e-7 < b

   1. Initial program 55.8%

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

      1. associate-/l* 55.9%

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

      2. +-commutative 55.9%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in a around 0 52.7%

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

      1. +-commutative 52.7%

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

      2. mul-1-neg 52.7%

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

      3. unsub-neg 52.7%

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

   7. Simplified 52.7%

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

      1. expm1-log1p-u 31.0%

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

      2. expm1-udef 12.9%

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

      3. clear-num 12.9%

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

      4. quot-tan 12.9%

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

   9. Applied egg-rr 12.9%

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

       1. expm1-def 31.1%

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

       2. expm1-log1p 52.7%

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

   11. Simplified 52.7%

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

   if -1.6e20 < b < 2.29999999999999995e-7

   1. Initial program 96.7%

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

      1. associate-/l* 96.6%

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

      2. remove-double-neg 96.6%

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

      3. sin-neg 96.6%

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

      4. neg-mul-1 96.6%

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

      5. associate-/r* 96.6%

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

      6. associate-/l* 96.7%

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

      7. *-commutative 96.7%

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

      8. associate-*l/ 96.8%

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

      9. associate-/l* 96.8%

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

      10. sin-neg 96.8%

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

      11. distribute-lft-neg-in 96.8%

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

      12. distribute-rgt-neg-in 96.8%

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

      13. associate-/l* 96.8%

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

      14. metadata-eval 96.8%

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

      15. /-rgt-identity 96.8%

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

      16. +-commutative 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in b around 0 96.7%

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

      1. *-commutative 96.7%

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

      2. clear-num 96.6%

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

      3. un-div-inv 96.6%

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

   7. Applied egg-rr 96.6%

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

      1. associate-/r/ 96.8%

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

   9. Simplified 96.8%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+20} \lor \neg \left(b \leq 2.3 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{r}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.6e+20) || !(b <= 9000.0)) {
		tmp = sin(b) * r;
	} 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 <= (-1.6d+20)) .or. (.not. (b <= 9000.0d0))) then
        tmp = sin(b) * r
    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 <= -1.6e+20) || !(b <= 9000.0)) {
		tmp = Math.sin(b) * r;
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 54.6%

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

      1. associate-/l* 54.6%

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

      2. remove-double-neg 54.6%

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

      3. sin-neg 54.6%

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

      4. neg-mul-1 54.6%

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

      5. associate-/r* 54.6%

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

      6. associate-/l* 54.6%

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

      7. *-commutative 54.6%

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

      8. associate-*l/ 54.5%

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

      9. associate-/l* 54.5%

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

      10. sin-neg 54.5%

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

      11. distribute-lft-neg-in 54.5%

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

      12. distribute-rgt-neg-in 54.5%

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

      13. associate-/l* 54.5%

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

      14. metadata-eval 54.5%

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

      15. /-rgt-identity 54.5%

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

      16. +-commutative 54.5%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in b around 0 7.0%

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

      1. mul-1-neg 7.0%

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

      2. unsub-neg 7.0%

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

   7. Simplified 7.0%

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

   8. Taylor expanded in a around 0 12.9%

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

      1. *-commutative 12.9%

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

   10. Simplified 12.9%

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

   if -1.6e20 < b < 9e3

   1. Initial program 96.8%

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

      1. associate-/l* 96.7%

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

      2. remove-double-neg 96.7%

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

      3. sin-neg 96.7%

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

      4. neg-mul-1 96.7%

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

      5. associate-/r* 96.7%

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

      6. associate-/l* 96.8%

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

      7. *-commutative 96.8%

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

      8. associate-*l/ 96.8%

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

      9. associate-/l* 96.8%

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

      10. sin-neg 96.8%

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

      11. distribute-lft-neg-in 96.8%

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

      12. distribute-rgt-neg-in 96.8%

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

      13. associate-/l* 96.8%

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

      14. metadata-eval 96.8%

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

      15. /-rgt-identity 96.8%

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

      16. +-commutative 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in b around 0 96.4%

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

4. Final simplification 62.8%

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

Alternative 12: 56.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+20} \lor \neg \left(b \leq 360000000\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.6e+20) (not (<= b 360000000.0)))
   (* (sin b) r)
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.6e+20) || !(b <= 360000000.0)) {
		tmp = sin(b) * r;
	} else {
		tmp = b * (r / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-1.6d+20)) .or. (.not. (b <= 360000000.0d0))) then
        tmp = sin(b) * r
    else
        tmp = b * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.6e+20) || !(b <= 360000000.0)) {
		tmp = Math.sin(b) * r;
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.6e+20) or not (b <= 360000000.0):
		tmp = math.sin(b) * r
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.6e+20) || !(b <= 360000000.0))
		tmp = Float64(sin(b) * r);
	else
		tmp = Float64(b * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -1.6e+20) || ~((b <= 360000000.0)))
		tmp = sin(b) * r;
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.6e+20], N[Not[LessEqual[b, 360000000.0]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.6e20 or 3.6e8 < b

   1. Initial program 54.6%

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

      1. associate-/l* 54.6%

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

      2. remove-double-neg 54.6%

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

      3. sin-neg 54.6%

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

      4. neg-mul-1 54.6%

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

      5. associate-/r* 54.6%

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

      6. associate-/l* 54.6%

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

      7. *-commutative 54.6%

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

      8. associate-*l/ 54.5%

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

      9. associate-/l* 54.5%

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

      10. sin-neg 54.5%

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

      11. distribute-lft-neg-in 54.5%

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

      12. distribute-rgt-neg-in 54.5%

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

      13. associate-/l* 54.5%

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

      14. metadata-eval 54.5%

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

      15. /-rgt-identity 54.5%

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

      16. +-commutative 54.5%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in b around 0 7.0%

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

      1. mul-1-neg 7.0%

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

      2. unsub-neg 7.0%

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

   7. Simplified 7.0%

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

   8. Taylor expanded in a around 0 12.9%

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

      1. *-commutative 12.9%

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

   10. Simplified 12.9%

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

   if -1.6e20 < b < 3.6e8

   1. Initial program 96.8%

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

      1. associate-/l* 96.7%

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

      2. remove-double-neg 96.7%

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

      3. sin-neg 96.7%

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

      4. neg-mul-1 96.7%

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

      5. associate-/r* 96.7%

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

      6. associate-/l* 96.8%

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

      7. *-commutative 96.8%

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

      8. associate-*l/ 96.8%

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

      9. associate-/l* 96.8%

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

      10. sin-neg 96.8%

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

      11. distribute-lft-neg-in 96.8%

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

      12. distribute-rgt-neg-in 96.8%

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

      13. associate-/l* 96.8%

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

      14. metadata-eval 96.8%

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

      15. /-rgt-identity 96.8%

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

      16. +-commutative 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in b around 0 96.4%

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

      1. *-commutative 96.4%

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

      2. clear-num 96.3%

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

      3. un-div-inv 96.2%

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

   7. Applied egg-rr 96.2%

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

      1. associate-/r/ 96.4%

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

   9. Simplified 96.4%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+20} \lor \neg \left(b \leq 360000000\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

   1. associate-/l* 79.7%

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

   2. remove-double-neg 79.7%

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

   3. sin-neg 79.7%

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

   4. neg-mul-1 79.7%

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

   5. associate-/r* 79.7%

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

   6. associate-/l* 79.8%

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

   7. *-commutative 79.8%

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

   8. associate-*l/ 79.8%

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

   9. associate-/l* 79.8%

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

   10. sin-neg 79.8%

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

   11. distribute-lft-neg-in 79.8%

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

   12. distribute-rgt-neg-in 79.8%

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

   13. associate-/l* 79.8%

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

   14. metadata-eval 79.8%

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

   15. /-rgt-identity 79.8%

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

   16. +-commutative 79.8%

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

3. Simplified 79.8%

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

5. Taylor expanded in b around 0 60.7%

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

   1. mul-1-neg 60.7%

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

   2. unsub-neg 60.7%

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

7. Simplified 60.7%

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

8. Taylor expanded in a around 0 42.9%

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

   1. *-commutative 42.9%

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

10. Simplified 42.9%

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

11. Final simplification 42.9%

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

Alternative 14: 35.4% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

   1. associate-/l* 79.7%

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

   2. remove-double-neg 79.7%

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

   3. sin-neg 79.7%

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

   4. neg-mul-1 79.7%

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

   5. associate-/r* 79.7%

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

   6. associate-/l* 79.8%

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

   7. *-commutative 79.8%

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

   8. associate-*l/ 79.8%

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

   9. associate-/l* 79.8%

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

   10. sin-neg 79.8%

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

   11. distribute-lft-neg-in 79.8%

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

   12. distribute-rgt-neg-in 79.8%

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

   13. associate-/l* 79.8%

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

   14. metadata-eval 79.8%

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

   15. /-rgt-identity 79.8%

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

   16. +-commutative 79.8%

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

3. Simplified 79.8%

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

5. Taylor expanded in b around 0 59.2%

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

6. Taylor expanded in a around 0 39.3%

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

7. Final simplification 39.3%

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

Reproduce

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