rsin B (should all be same)

Percentage Accurate: 76.9% → 99.5%

Time: 18.9s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

   1. associate-*r/ 76.7%

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

   2. +-commutative 76.7%

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

3. Simplified 76.7%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-define 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

   \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)} \]
8. 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 76.7%

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

   1. +-commutative 76.7%

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

3. Simplified 76.7%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

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(Float64(r * 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[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

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

   1. associate-*r/ 76.7%

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

   2. +-commutative 76.7%

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

3. Simplified 76.7%

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

   1. cos-sum 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 4: 77.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

   1. associate-*r/ 76.7%

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

   2. +-commutative 76.7%

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

3. Simplified 76.7%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-define 99.5%

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

6. Applied egg-rr 99.5%

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

   1. distribute-lft-neg-out 99.5%

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

   2. fmm-def 99.5%

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

   3. add-sqr-sqrt 49.4%

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

   4. sqrt-unprod 87.1%

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

   5. sqr-neg 87.1%

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

   6. sqrt-unprod 37.7%

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

   7. add-sqr-sqrt 76.6%

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

   8. neg-mul-1 76.6%

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

   9. associate-*l* 76.6%

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

   10. add-sqr-sqrt 38.9%

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

   11. sqrt-unprod 89.0%

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

   12. sqr-neg 89.0%

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

   13. sqrt-unprod 50.1%

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

   14. add-sqr-sqrt 99.5%

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

8. Applied egg-rr 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-undefine 99.5%

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

   3. *-commutative 99.5%

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

   4. neg-mul-1 99.5%

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

   5. *-commutative 99.5%

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

   6. neg-mul-1 99.5%

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

   7. distribute-neg-out 99.5%

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

   8. fma-undefine 99.5%

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

   9. fmm-undef 99.5%

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

   10. *-commutative 99.5%

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

   11. neg-mul-1 99.5%

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

   12. distribute-lft-neg-in 99.5%

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

   13. cancel-sign-sub 99.5%

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

   14. *-commutative 99.5%

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

10. Simplified 99.5%

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

12. Applied egg-rr 77.7%

    \[\leadsto \color{blue}{\left(\sin b \cdot \left(-r\right)\right) \cdot \frac{1}{\left(\sin b \cdot \sin a\right) \cdot 2 - \cos \left(b - a\right)}} \]
13. Step-by-step derivation

    1. associate-*r/ 77.8%

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

    2. *-rgt-identity 77.8%

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

    3. distribute-rgt-neg-out 77.8%

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

    4. distribute-neg-frac 77.8%

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

    5. distribute-neg-frac2 77.8%

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

    6. neg-sub0 77.8%

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

    7. associate--r- 77.8%

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

    8. neg-sub0 77.8%

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

    9. distribute-lft-neg-in 77.8%

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

    10. distribute-lft-neg-in 77.8%

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

    11. *-commutative 77.8%

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

    12. associate-*l* 77.8%

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

    13. distribute-lft-neg-in 77.8%

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

    14. distribute-rgt-neg-in 77.8%

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

    15. metadata-eval 77.8%

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

14. Simplified 77.8%

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

15. Final simplification 77.8%

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

Alternative 5: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -200000 \lor \neg \left(a \leq 9.8 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -200000.0) (not (<= a 9.8e-6)))
   (* r (/ (sin b) (cos a)))
   (* r (/ (sin b) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -200000.0) || !(a <= 9.8e-6)) {
		tmp = r * (sin(b) / cos(a));
	} else {
		tmp = r * (sin(b) / cos(b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-200000.0d0)) .or. (.not. (a <= 9.8d-6))) then
        tmp = r * (sin(b) / cos(a))
    else
        tmp = r * (sin(b) / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -200000.0) || !(a <= 9.8e-6)) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -200000.0) or not (a <= 9.8e-6):
		tmp = r * (math.sin(b) / math.cos(a))
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -200000.0) || !(a <= 9.8e-6))
		tmp = Float64(r * Float64(sin(b) / 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 ((a <= -200000.0) || ~((a <= 9.8e-6)))
		tmp = r * (sin(b) / cos(a));
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -200000.0], N[Not[LessEqual[a, 9.8e-6]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2e5 or 9.79999999999999934e-6 < a

   1. Initial program 57.6%

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

      1. +-commutative 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in b around 0 58.4%

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

   if -2e5 < a < 9.79999999999999934e-6

   1. Initial program 97.9%

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

      1. +-commutative 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in a around 0 97.9%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -200000 \lor \neg \left(a \leq 9.8 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -200000 \lor \neg \left(a \leq 4.9 \cdot 10^{-5}\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -200000.0) (not (<= a 4.9e-5)))
   (* (sin b) (/ r (cos a)))
   (* r (/ (sin b) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -200000.0) || !(a <= 4.9e-5)) {
		tmp = sin(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 ((a <= (-200000.0d0)) .or. (.not. (a <= 4.9d-5))) then
        tmp = sin(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 ((a <= -200000.0) || !(a <= 4.9e-5)) {
		tmp = Math.sin(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 (a <= -200000.0) or not (a <= 4.9e-5):
		tmp = math.sin(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 ((a <= -200000.0) || !(a <= 4.9e-5))
		tmp = Float64(sin(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 ((a <= -200000.0) || ~((a <= 4.9e-5)))
		tmp = sin(b) * (r / cos(a));
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -200000.0], N[Not[LessEqual[a, 4.9e-5]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] * 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 2 regimes

2. if a < -2e5 or 4.9e-5 < a

   1. Initial program 57.6%

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

      1. +-commutative 57.6%

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

   3. Simplified 57.6%

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

      1. add-cbrt-cube 34.9%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)}} \]

      2. pow1/3 26.4%

         \[\leadsto \color{blue}{{\left(\left(\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right)}^{0.3333333333333333}} \]

      3. pow3 26.4%

         \[\leadsto {\color{blue}{\left({\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. associate-*r/ 26.4%

         \[\leadsto {\left({\color{blue}{\left(\frac{r \cdot \sin b}{\cos \left(b + a\right)}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. *-commutative 26.4%

         \[\leadsto {\left({\left(\frac{\color{blue}{\sin b \cdot r}}{\cos \left(b + a\right)}\right)}^{3}\right)}^{0.3333333333333333} \]

      6. associate-/l* 26.4%

         \[\leadsto {\left({\color{blue}{\left(\sin b \cdot \frac{r}{\cos \left(b + a\right)}\right)}}^{3}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 26.4%

      \[\leadsto \color{blue}{{\left({\left(\sin b \cdot \frac{r}{\cos \left(b + a\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in b around 0 26.3%

      \[\leadsto {\left({\left(\sin b \cdot \frac{r}{\color{blue}{\cos a}}\right)}^{3}\right)}^{0.3333333333333333} \]
   8. Step-by-step derivation

      1. unpow1/3 35.0%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(\sin b \cdot \frac{r}{\cos a}\right)}^{3}}} \]

      2. rem-cbrt-cube 58.4%

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

      3. *-commutative 58.4%

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

   9. Applied egg-rr 58.4%

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

   if -2e5 < a < 4.9e-5

   1. Initial program 97.9%

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

      1. +-commutative 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in a around 0 97.9%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -200000 \lor \neg \left(a \leq 4.9 \cdot 10^{-5}\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -200000 \lor \neg \left(a \leq 4.9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -200000.0) (not (<= a 4.9e-5)))
   (/ (sin b) (/ (cos a) r))
   (* r (/ (sin b) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -200000.0) || !(a <= 4.9e-5)) {
		tmp = sin(b) / (cos(a) / r);
	} 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 ((a <= (-200000.0d0)) .or. (.not. (a <= 4.9d-5))) then
        tmp = sin(b) / (cos(a) / r)
    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 ((a <= -200000.0) || !(a <= 4.9e-5)) {
		tmp = Math.sin(b) / (Math.cos(a) / r);
	} else {
		tmp = r * (Math.sin(b) / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -200000.0) or not (a <= 4.9e-5):
		tmp = math.sin(b) / (math.cos(a) / r)
	else:
		tmp = r * (math.sin(b) / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -200000.0) || !(a <= 4.9e-5))
		tmp = Float64(sin(b) / Float64(cos(a) / r));
	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 ((a <= -200000.0) || ~((a <= 4.9e-5)))
		tmp = sin(b) / (cos(a) / r);
	else
		tmp = r * (sin(b) / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -200000.0], N[Not[LessEqual[a, 4.9e-5]], $MachinePrecision]], N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[a], $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2e5 or 4.9e-5 < a

   1. Initial program 57.6%

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

      1. +-commutative 57.6%

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

   3. Simplified 57.6%

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

      1. add-cbrt-cube 34.9%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)}} \]

      2. pow1/3 26.4%

         \[\leadsto \color{blue}{{\left(\left(\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right)}^{0.3333333333333333}} \]

      3. pow3 26.4%

         \[\leadsto {\color{blue}{\left({\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. associate-*r/ 26.4%

         \[\leadsto {\left({\color{blue}{\left(\frac{r \cdot \sin b}{\cos \left(b + a\right)}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. *-commutative 26.4%

         \[\leadsto {\left({\left(\frac{\color{blue}{\sin b \cdot r}}{\cos \left(b + a\right)}\right)}^{3}\right)}^{0.3333333333333333} \]

      6. associate-/l* 26.4%

         \[\leadsto {\left({\color{blue}{\left(\sin b \cdot \frac{r}{\cos \left(b + a\right)}\right)}}^{3}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 26.4%

      \[\leadsto \color{blue}{{\left({\left(\sin b \cdot \frac{r}{\cos \left(b + a\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in b around 0 26.3%

      \[\leadsto {\left({\left(\sin b \cdot \frac{r}{\color{blue}{\cos a}}\right)}^{3}\right)}^{0.3333333333333333} \]
   8. Step-by-step derivation

      1. unpow1/3 35.0%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(\sin b \cdot \frac{r}{\cos a}\right)}^{3}}} \]

      2. rem-cbrt-cube 58.4%

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

      3. clear-num 58.4%

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

      4. un-div-inv 58.4%

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

   9. Applied egg-rr 58.4%

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

   if -2e5 < a < 4.9e-5

   1. Initial program 97.9%

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

      1. +-commutative 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in a around 0 97.9%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -200000 \lor \neg \left(a \leq 4.9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -550000000000:\\ \;\;\;\;\left|t\_0\right|\\ \mathbf{elif}\;b \leq 7800:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -550000000000.0)
     (fabs t_0)
     (if (<= b 7800.0) (/ (* r b) (cos (+ b a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -550000000000.0) {
		tmp = fabs(t_0);
	} else if (b <= 7800.0) {
		tmp = (r * b) / cos((b + a));
	} else {
		tmp = 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 = r * sin(b)
    if (b <= (-550000000000.0d0)) then
        tmp = abs(t_0)
    else if (b <= 7800.0d0) then
        tmp = (r * b) / cos((b + a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.sin(b);
	double tmp;
	if (b <= -550000000000.0) {
		tmp = Math.abs(t_0);
	} else if (b <= 7800.0) {
		tmp = (r * b) / Math.cos((b + a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.sin(b)
	tmp = 0
	if b <= -550000000000.0:
		tmp = math.fabs(t_0)
	elif b <= 7800.0:
		tmp = (r * b) / math.cos((b + a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -550000000000.0)
		tmp = abs(t_0);
	elseif (b <= 7800.0)
		tmp = Float64(Float64(r * b) / cos(Float64(b + a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * sin(b);
	tmp = 0.0;
	if (b <= -550000000000.0)
		tmp = abs(t_0);
	elseif (b <= 7800.0)
		tmp = (r * b) / cos((b + a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -550000000000.0], N[Abs[t$95$0], $MachinePrecision], If[LessEqual[b, 7800.0], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.5e11

   1. Initial program 52.4%

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

      1. +-commutative 52.4%

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

   3. Simplified 52.4%

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

      1. add-cbrt-cube 18.3%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)}} \]

      2. pow1/3 11.0%

         \[\leadsto \color{blue}{{\left(\left(\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right)}^{0.3333333333333333}} \]

      3. pow3 11.0%

         \[\leadsto {\color{blue}{\left({\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. associate-*r/ 11.0%

         \[\leadsto {\left({\color{blue}{\left(\frac{r \cdot \sin b}{\cos \left(b + a\right)}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. *-commutative 11.0%

         \[\leadsto {\left({\left(\frac{\color{blue}{\sin b \cdot r}}{\cos \left(b + a\right)}\right)}^{3}\right)}^{0.3333333333333333} \]

      6. associate-/l* 11.0%

         \[\leadsto {\left({\color{blue}{\left(\sin b \cdot \frac{r}{\cos \left(b + a\right)}\right)}}^{3}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 11.0%

      \[\leadsto \color{blue}{{\left({\left(\sin b \cdot \frac{r}{\cos \left(b + a\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in b around 0 3.8%

      \[\leadsto {\left({\left(\sin b \cdot \frac{r}{\color{blue}{\cos a}}\right)}^{3}\right)}^{0.3333333333333333} \]

   8. Taylor expanded in a around 0 10.9%

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

      1. add-sqr-sqrt 5.3%

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

      2. sqrt-unprod 7.7%

         \[\leadsto \color{blue}{\sqrt{\left(r \cdot \sin b\right) \cdot \left(r \cdot \sin b\right)}} \]

      3. pow2 7.7%

         \[\leadsto \sqrt{\color{blue}{{\left(r \cdot \sin b\right)}^{2}}} \]

      4. *-commutative 7.7%

         \[\leadsto \sqrt{{\color{blue}{\left(\sin b \cdot r\right)}}^{2}} \]

   10. Applied egg-rr 7.7%

       \[\leadsto \color{blue}{\sqrt{{\left(\sin b \cdot r\right)}^{2}}} \]
   11. Step-by-step derivation

       1. unpow2 7.7%

          \[\leadsto \sqrt{\color{blue}{\left(\sin b \cdot r\right) \cdot \left(\sin b \cdot r\right)}} \]

       2. rem-sqrt-square 11.1%

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

   12. Simplified 11.1%

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

   if -5.5e11 < b < 7800

   1. Initial program 98.3%

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

      1. associate-*r/ 98.4%

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

      2. +-commutative 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in b around 0 97.4%

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

   if 7800 < b

   1. Initial program 56.3%

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

      1. +-commutative 56.3%

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

   3. Simplified 56.3%

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

      1. add-cbrt-cube 24.2%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)}} \]

      2. pow1/3 17.6%

         \[\leadsto \color{blue}{{\left(\left(\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right)}^{0.3333333333333333}} \]

      3. pow3 17.6%

         \[\leadsto {\color{blue}{\left({\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. associate-*r/ 17.6%

         \[\leadsto {\left({\color{blue}{\left(\frac{r \cdot \sin b}{\cos \left(b + a\right)}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. *-commutative 17.6%

         \[\leadsto {\left({\left(\frac{\color{blue}{\sin b \cdot r}}{\cos \left(b + a\right)}\right)}^{3}\right)}^{0.3333333333333333} \]

      6. associate-/l* 17.6%

         \[\leadsto {\left({\color{blue}{\left(\sin b \cdot \frac{r}{\cos \left(b + a\right)}\right)}}^{3}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 17.6%

      \[\leadsto \color{blue}{{\left({\left(\sin b \cdot \frac{r}{\cos \left(b + a\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in b around 0 5.8%

      \[\leadsto {\left({\left(\sin b \cdot \frac{r}{\color{blue}{\cos a}}\right)}^{3}\right)}^{0.3333333333333333} \]

   8. Taylor expanded in a around 0 11.0%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -550000000000:\\ \;\;\;\;\left|r \cdot \sin b\right|\\ \mathbf{elif}\;b \leq 7800:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

3. Final simplification 76.7%

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

Alternative 10: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

   1. associate-*r/ 76.7%

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

   2. +-commutative 76.7%

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

3. Simplified 76.7%

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

   1. *-commutative 76.7%

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

   2. associate-/l* 76.7%

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

6. Applied egg-rr 76.7%

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

7. Final simplification 76.7%

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

Alternative 11: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

   1. associate-*r/ 76.7%

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

   2. +-commutative 76.7%

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

3. Simplified 76.7%

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

5. Final simplification 76.7%

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

Alternative 12: 54.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

   1. +-commutative 76.7%

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

3. Simplified 76.7%

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

5. Taylor expanded in b around 0 55.2%

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

6. Final simplification 55.2%

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

Alternative 13: 54.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8500000 \lor \neg \left(b \leq 7800\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -8500000.0) (not (<= b 7800.0)))
   (* r (sin b))
   (* b (/ r (cos (+ b a))))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -8500000.0) || !(b <= 7800.0)) {
		tmp = r * sin(b);
	} else {
		tmp = b * (r / cos((b + a)));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-8500000.0d0)) .or. (.not. (b <= 7800.0d0))) then
        tmp = r * sin(b)
    else
        tmp = b * (r / cos((b + a)))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -8500000.0) || !(b <= 7800.0)) {
		tmp = r * Math.sin(b);
	} else {
		tmp = b * (r / Math.cos((b + a)));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -8500000.0) or not (b <= 7800.0):
		tmp = r * math.sin(b)
	else:
		tmp = b * (r / math.cos((b + a)))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -8500000.0) || !(b <= 7800.0))
		tmp = Float64(r * sin(b));
	else
		tmp = Float64(b * Float64(r / cos(Float64(b + a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -8500000.0) || ~((b <= 7800.0)))
		tmp = r * sin(b);
	else
		tmp = b * (r / cos((b + a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -8500000.0], N[Not[LessEqual[b, 7800.0]], $MachinePrecision]], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -8.5e6 or 7800 < b

   1. Initial program 54.1%

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

      1. +-commutative 54.1%

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

   3. Simplified 54.1%

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

      1. add-cbrt-cube 21.2%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)}} \]

      2. pow1/3 14.2%

         \[\leadsto \color{blue}{{\left(\left(\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right)}^{0.3333333333333333}} \]

      3. pow3 14.2%

         \[\leadsto {\color{blue}{\left({\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. associate-*r/ 14.2%

         \[\leadsto {\left({\color{blue}{\left(\frac{r \cdot \sin b}{\cos \left(b + a\right)}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. *-commutative 14.2%

         \[\leadsto {\left({\left(\frac{\color{blue}{\sin b \cdot r}}{\cos \left(b + a\right)}\right)}^{3}\right)}^{0.3333333333333333} \]

      6. associate-/l* 14.2%

         \[\leadsto {\left({\color{blue}{\left(\sin b \cdot \frac{r}{\cos \left(b + a\right)}\right)}}^{3}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 14.2%

      \[\leadsto \color{blue}{{\left({\left(\sin b \cdot \frac{r}{\cos \left(b + a\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in b around 0 4.8%

      \[\leadsto {\left({\left(\sin b \cdot \frac{r}{\color{blue}{\cos a}}\right)}^{3}\right)}^{0.3333333333333333} \]

   8. Taylor expanded in a around 0 11.1%

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

   if -8.5e6 < b < 7800

   1. Initial program 98.9%

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

      1. associate-*r/ 98.9%

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

      2. +-commutative 98.9%

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

   3. Simplified 98.9%

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

      1. add-exp-log 59.1%

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

      2. *-commutative 59.1%

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

   6. Applied egg-rr 59.1%

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

      1. add-sqr-sqrt 42.1%

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

      2. sqrt-unprod 50.9%

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

      3. pow2 50.9%

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

   8. Applied egg-rr 50.9%

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

      1. unpow2 50.9%

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

      2. rem-sqrt-square 50.9%

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

      3. +-commutative 50.9%

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

   10. Simplified 50.9%

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

   11. Taylor expanded in b around 0 79.3%

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

       1. +-commutative 79.3%

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

       2. rem-square-sqrt 70.2%

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

       3. fabs-sqr 70.2%

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

       4. rem-square-sqrt 98.1%

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

       5. associate-*r/ 98.1%

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

   13. Simplified 98.1%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8500000 \lor \neg \left(b \leq 7800\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos \left(b + a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -23000 \lor \neg \left(b \leq 4.5\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -23000.0) (not (<= b 4.5)))
   (* r (sin b))
   (/ (* r b) (cos (+ b a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -23000.0) || !(b <= 4.5)) {
		tmp = r * sin(b);
	} else {
		tmp = (r * b) / cos((b + a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-23000.0d0)) .or. (.not. (b <= 4.5d0))) then
        tmp = r * sin(b)
    else
        tmp = (r * b) / cos((b + a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -23000.0) || !(b <= 4.5)) {
		tmp = r * Math.sin(b);
	} else {
		tmp = (r * b) / Math.cos((b + a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -23000.0) or not (b <= 4.5):
		tmp = r * math.sin(b)
	else:
		tmp = (r * b) / math.cos((b + a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -23000.0) || !(b <= 4.5))
		tmp = Float64(r * sin(b));
	else
		tmp = Float64(Float64(r * b) / cos(Float64(b + a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -23000.0) || ~((b <= 4.5)))
		tmp = r * sin(b);
	else
		tmp = (r * b) / cos((b + a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -23000.0], N[Not[LessEqual[b, 4.5]], $MachinePrecision]], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(N[(r * b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -23000 or 4.5 < b

   1. Initial program 54.1%

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

      1. +-commutative 54.1%

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

   3. Simplified 54.1%

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

      1. add-cbrt-cube 21.2%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)}} \]

      2. pow1/3 14.2%

         \[\leadsto \color{blue}{{\left(\left(\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right)}^{0.3333333333333333}} \]

      3. pow3 14.2%

         \[\leadsto {\color{blue}{\left({\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. associate-*r/ 14.2%

         \[\leadsto {\left({\color{blue}{\left(\frac{r \cdot \sin b}{\cos \left(b + a\right)}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. *-commutative 14.2%

         \[\leadsto {\left({\left(\frac{\color{blue}{\sin b \cdot r}}{\cos \left(b + a\right)}\right)}^{3}\right)}^{0.3333333333333333} \]

      6. associate-/l* 14.2%

         \[\leadsto {\left({\color{blue}{\left(\sin b \cdot \frac{r}{\cos \left(b + a\right)}\right)}}^{3}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 14.2%

      \[\leadsto \color{blue}{{\left({\left(\sin b \cdot \frac{r}{\cos \left(b + a\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in b around 0 4.8%

      \[\leadsto {\left({\left(\sin b \cdot \frac{r}{\color{blue}{\cos a}}\right)}^{3}\right)}^{0.3333333333333333} \]

   8. Taylor expanded in a around 0 11.1%

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

   if -23000 < b < 4.5

   1. Initial program 98.9%

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

      1. associate-*r/ 98.9%

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

      2. +-commutative 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in b around 0 98.1%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -23000 \lor \neg \left(b \leq 4.5\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.8 \lor \neg \left(b \leq 1\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -4.8) (not (<= b 1.0))) (* r (sin b)) (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -4.8) || !(b <= 1.0)) {
		tmp = r * sin(b);
	} else {
		tmp = b * (r / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-4.8d0)) .or. (.not. (b <= 1.0d0))) then
        tmp = r * sin(b)
    else
        tmp = b * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -4.8) || !(b <= 1.0)) {
		tmp = r * Math.sin(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -4.8) or not (b <= 1.0):
		tmp = r * math.sin(b)
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -4.8) || !(b <= 1.0))
		tmp = Float64(r * sin(b));
	else
		tmp = Float64(b * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -4.8) || ~((b <= 1.0)))
		tmp = r * sin(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -4.8], N[Not[LessEqual[b, 1.0]], $MachinePrecision]], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.79999999999999982 or 1 < b

   1. Initial program 54.4%

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

      1. +-commutative 54.4%

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

   3. Simplified 54.4%

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

      1. add-cbrt-cube 21.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)}} \]

      2. pow1/3 14.2%

         \[\leadsto \color{blue}{{\left(\left(\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right)}^{0.3333333333333333}} \]

      3. pow3 14.2%

         \[\leadsto {\color{blue}{\left({\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. associate-*r/ 14.2%

         \[\leadsto {\left({\color{blue}{\left(\frac{r \cdot \sin b}{\cos \left(b + a\right)}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. *-commutative 14.2%

         \[\leadsto {\left({\left(\frac{\color{blue}{\sin b \cdot r}}{\cos \left(b + a\right)}\right)}^{3}\right)}^{0.3333333333333333} \]

      6. associate-/l* 14.2%

         \[\leadsto {\left({\color{blue}{\left(\sin b \cdot \frac{r}{\cos \left(b + a\right)}\right)}}^{3}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 14.2%

      \[\leadsto \color{blue}{{\left({\left(\sin b \cdot \frac{r}{\cos \left(b + a\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in b around 0 4.9%

      \[\leadsto {\left({\left(\sin b \cdot \frac{r}{\color{blue}{\cos a}}\right)}^{3}\right)}^{0.3333333333333333} \]

   8. Taylor expanded in a around 0 11.0%

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

   if -4.79999999999999982 < b < 1

   1. Initial program 98.9%

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

      1. +-commutative 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in b around 0 98.7%

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

      1. associate-/l* 98.7%

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

   7. Simplified 98.7%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.8 \lor \neg \left(b \leq 1\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 54.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.8 \lor \neg \left(b \leq 1.55\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -4.8) (not (<= b 1.55))) (* r (sin b)) (/ (* r b) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -4.8) || !(b <= 1.55)) {
		tmp = r * sin(b);
	} else {
		tmp = (r * b) / cos(a);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-4.8d0)) .or. (.not. (b <= 1.55d0))) then
        tmp = r * sin(b)
    else
        tmp = (r * b) / cos(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -4.8) || !(b <= 1.55)) {
		tmp = r * Math.sin(b);
	} else {
		tmp = (r * b) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -4.8) or not (b <= 1.55):
		tmp = r * math.sin(b)
	else:
		tmp = (r * b) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -4.8) || !(b <= 1.55))
		tmp = Float64(r * sin(b));
	else
		tmp = Float64(Float64(r * b) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -4.8) || ~((b <= 1.55)))
		tmp = r * sin(b);
	else
		tmp = (r * b) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -4.8], N[Not[LessEqual[b, 1.55]], $MachinePrecision]], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.79999999999999982 or 1.55000000000000004 < b

   1. Initial program 54.4%

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

      1. +-commutative 54.4%

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

   3. Simplified 54.4%

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

      1. add-cbrt-cube 21.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)}} \]

      2. pow1/3 14.2%

         \[\leadsto \color{blue}{{\left(\left(\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right)}^{0.3333333333333333}} \]

      3. pow3 14.2%

         \[\leadsto {\color{blue}{\left({\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. associate-*r/ 14.2%

         \[\leadsto {\left({\color{blue}{\left(\frac{r \cdot \sin b}{\cos \left(b + a\right)}\right)}}^{3}\right)}^{0.3333333333333333} \]

      5. *-commutative 14.2%

         \[\leadsto {\left({\left(\frac{\color{blue}{\sin b \cdot r}}{\cos \left(b + a\right)}\right)}^{3}\right)}^{0.3333333333333333} \]

      6. associate-/l* 14.2%

         \[\leadsto {\left({\color{blue}{\left(\sin b \cdot \frac{r}{\cos \left(b + a\right)}\right)}}^{3}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 14.2%

      \[\leadsto \color{blue}{{\left({\left(\sin b \cdot \frac{r}{\cos \left(b + a\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. Taylor expanded in b around 0 4.9%

      \[\leadsto {\left({\left(\sin b \cdot \frac{r}{\color{blue}{\cos a}}\right)}^{3}\right)}^{0.3333333333333333} \]

   8. Taylor expanded in a around 0 11.0%

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

   if -4.79999999999999982 < b < 1.55000000000000004

   1. Initial program 98.9%

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

      1. +-commutative 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in b around 0 98.7%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.8 \lor \neg \left(b \leq 1.55\right):\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 38.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

   1. +-commutative 76.7%

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

3. Simplified 76.7%

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

   1. add-cbrt-cube 40.2%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)}} \]

   2. pow1/3 29.7%

      \[\leadsto \color{blue}{{\left(\left(\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right) \cdot \left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)\right)}^{0.3333333333333333}} \]

   3. pow3 29.7%

      \[\leadsto {\color{blue}{\left({\left(r \cdot \frac{\sin b}{\cos \left(b + a\right)}\right)}^{3}\right)}}^{0.3333333333333333} \]

   4. associate-*r/ 29.7%

      \[\leadsto {\left({\color{blue}{\left(\frac{r \cdot \sin b}{\cos \left(b + a\right)}\right)}}^{3}\right)}^{0.3333333333333333} \]

   5. *-commutative 29.7%

      \[\leadsto {\left({\left(\frac{\color{blue}{\sin b \cdot r}}{\cos \left(b + a\right)}\right)}^{3}\right)}^{0.3333333333333333} \]

   6. associate-/l* 29.7%

      \[\leadsto {\left({\color{blue}{\left(\sin b \cdot \frac{r}{\cos \left(b + a\right)}\right)}}^{3}\right)}^{0.3333333333333333} \]

6. Applied egg-rr 29.7%

   \[\leadsto \color{blue}{{\left({\left(\sin b \cdot \frac{r}{\cos \left(b + a\right)}\right)}^{3}\right)}^{0.3333333333333333}} \]

7. Taylor expanded in b around 0 25.0%

   \[\leadsto {\left({\left(\sin b \cdot \frac{r}{\color{blue}{\cos a}}\right)}^{3}\right)}^{0.3333333333333333} \]

8. Taylor expanded in a around 0 36.7%

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

9. Final simplification 36.7%

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

Alternative 18: 34.4% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

   1. +-commutative 76.7%

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

3. Simplified 76.7%

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

5. Taylor expanded in b around 0 51.0%

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

   1. associate-/l* 51.0%

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

7. Simplified 51.0%

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

8. Taylor expanded in a around 0 32.9%

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

   1. *-commutative 32.9%

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

10. Simplified 32.9%

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

11. Final simplification 32.9%

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

Reproduce

herbie shell --seed 2024112 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))