rsin A (should all be same)

Percentage Accurate: 76.3% → 99.5%

Time: 19.3s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.3% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot \sin a\\ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, t\_0\right) - \mathsf{fma}\left(\sin b, \sin a, t\_0\right)} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. associate-/l* 77.2%

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

   2. +-commutative 77.2%

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

3. Simplified 77.2%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-define 99.6%

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

6. Applied egg-rr 99.6%

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

   1. distribute-lft-neg-out 99.6%

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

   2. fma-neg 99.5%

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

   3. add-sqr-sqrt 50.1%

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

   4. sqrt-unprod 88.6%

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

   5. sqr-neg 88.6%

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

   6. sqrt-unprod 38.5%

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

   7. add-sqr-sqrt 77.5%

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

   8. distribute-lft-neg-out 77.5%

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

   9. neg-mul-1 77.5%

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

   10. add-sqr-sqrt 39.0%

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

   11. sqrt-unprod 88.4%

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

   12. sqr-neg 88.4%

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

   13. sqrt-unprod 49.3%

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

   14. add-sqr-sqrt 99.5%

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

   15. prod-diff 99.6%

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

8. Applied egg-rr 99.6%

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

   1. +-commutative 99.6%

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

   2. fma-undefine 99.6%

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

   3. *-commutative 99.6%

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

   4. neg-mul-1 99.6%

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

   5. *-commutative 99.6%

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

   6. neg-mul-1 99.6%

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

   7. distribute-neg-out 99.6%

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

   8. fma-undefine 99.6%

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

   9. *-commutative 99.6%

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

   10. fma-undefine 99.6%

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

   11. *-commutative 99.6%

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

   12. fma-undefine 99.6%

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

   13. *-commutative 99.6%

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

   14. neg-mul-1 99.6%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. associate-/l* 77.2%

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

   2. +-commutative 77.2%

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

3. Simplified 77.2%

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

   1. cos-sum 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. fma-define 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. associate-/l* 77.2%

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

   2. +-commutative 77.2%

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

3. Simplified 77.2%

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

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

Alternative 4: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\cos a}{r} \cdot \frac{\cos b}{\sin b} - \frac{\sin a}{r}} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ 1.0 (- (* (/ (cos a) r) (/ (cos b) (sin b))) (/ (sin a) r))))

C

double code(double r, double a, double b) {
	return 1.0 / (((cos(a) / r) * (cos(b) / sin(b))) - (sin(a) / 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 = 1.0d0 / (((cos(a) / r) * (cos(b) / sin(b))) - (sin(a) / r))
end function

Java

public static double code(double r, double a, double b) {
	return 1.0 / (((Math.cos(a) / r) * (Math.cos(b) / Math.sin(b))) - (Math.sin(a) / r));
}

Python

def code(r, a, b):
	return 1.0 / (((math.cos(a) / r) * (math.cos(b) / math.sin(b))) - (math.sin(a) / r))

Julia

function code(r, a, b)
	return Float64(1.0 / Float64(Float64(Float64(cos(a) / r) * Float64(cos(b) / sin(b))) - Float64(sin(a) / r)))
end

MATLAB

function tmp = code(r, a, b)
	tmp = 1.0 / (((cos(a) / r) * (cos(b) / sin(b))) - (sin(a) / r));
end

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. associate-/l* 77.2%

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

   2. +-commutative 77.2%

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

3. Simplified 77.2%

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

   1. associate-*r/ 77.2%

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

   2. clear-num 76.4%

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

   3. *-commutative 76.4%

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

6. Applied egg-rr 76.4%

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

   1. *-commutative 76.4%

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

   2. cos-sum 98.6%

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

   3. div-sub 91.3%

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

   4. *-commutative 91.3%

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

   5. *-commutative 91.3%

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

8. Applied egg-rr 91.3%

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

   1. *-commutative 91.3%

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

   2. *-commutative 91.3%

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

   3. times-frac 91.1%

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

   4. remove-double-neg 91.1%

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

   5. neg-mul-1 91.1%

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

   6. *-commutative 91.1%

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

   7. distribute-rgt-neg-in 91.1%

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

   8. distribute-frac-neg 91.1%

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

   9. associate-*r* 91.1%

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

   10. *-commutative 91.1%

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

   11. neg-mul-1 91.1%

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

   12. distribute-neg-frac 91.1%

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

   13. times-frac 98.5%

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

   14. *-inverses 98.5%

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

   15. distribute-lft-neg-in 98.5%

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

   16. metadata-eval 98.5%

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

   17. associate-*r/ 98.5%

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

10. Simplified 98.5%

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

Alternative 5: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-6} \lor \neg \left(b \leq 0.000135\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -1.6e-6) (not (<= b 0.000135)))
   (* r (/ (sin b) (cos b)))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.6e-6) || !(b <= 0.000135)) {
		tmp = r * (sin(b) / cos(b));
	} else {
		tmp = b * (r / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-1.6d-6)) .or. (.not. (b <= 0.000135d0))) then
        tmp = r * (sin(b) / cos(b))
    else
        tmp = b * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -1.6e-6) || !(b <= 0.000135)) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -1.6e-6) or not (b <= 0.000135):
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -1.6e-6) || !(b <= 0.000135))
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = Float64(b * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -1.6e-6) || ~((b <= 0.000135)))
		tmp = r * (sin(b) / cos(b));
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -1.6e-6], N[Not[LessEqual[b, 0.000135]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.5999999999999999e-6 or 1.35000000000000002e-4 < b

   1. Initial program 52.5%

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

      1. associate-/l* 52.6%

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

      2. +-commutative 52.6%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in a around 0 53.4%

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

   if -1.5999999999999999e-6 < b < 1.35000000000000002e-4

   1. Initial program 98.7%

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

      1. associate-/l* 98.7%

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

      2. +-commutative 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in b around 0 98.7%

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

      1. clear-num 98.6%

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

      2. un-div-inv 98.5%

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

   7. Applied egg-rr 98.5%

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

      1. associate-/r/ 98.7%

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

   9. Simplified 98.7%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-6} \lor \neg \left(b \leq 0.000135\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 0.00052:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -4.2e-6)
   (* r (/ (sin b) (cos b)))
   (if (<= b 0.00052) (* b (/ r (cos a))) (* (sin b) (/ r (cos b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -4.2e-6) {
		tmp = r * (sin(b) / cos(b));
	} else if (b <= 0.00052) {
		tmp = b * (r / cos(a));
	} else {
		tmp = sin(b) * (r / cos(b));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-4.2d-6)) then
        tmp = r * (sin(b) / cos(b))
    else if (b <= 0.00052d0) then
        tmp = b * (r / cos(a))
    else
        tmp = sin(b) * (r / cos(b))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -4.2e-6) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else if (b <= 0.00052) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = Math.sin(b) * (r / Math.cos(b));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -4.2e-6:
		tmp = r * (math.sin(b) / math.cos(b))
	elif b <= 0.00052:
		tmp = b * (r / math.cos(a))
	else:
		tmp = math.sin(b) * (r / math.cos(b))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -4.2e-6)
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	elseif (b <= 0.00052)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -4.2e-6)
		tmp = r * (sin(b) / cos(b));
	elseif (b <= 0.00052)
		tmp = b * (r / cos(a));
	else
		tmp = sin(b) * (r / cos(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -4.2e-6], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.00052], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.1999999999999996e-6

   1. Initial program 51.8%

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

      1. associate-/l* 51.9%

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

      2. +-commutative 51.9%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in a around 0 52.0%

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

   if -4.1999999999999996e-6 < b < 5.19999999999999954e-4

   1. Initial program 98.7%

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

      1. associate-/l* 98.7%

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

      2. +-commutative 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in b around 0 98.7%

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

      1. clear-num 98.6%

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

      2. un-div-inv 98.5%

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

   7. Applied egg-rr 98.5%

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

      1. associate-/r/ 98.7%

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

   9. Simplified 98.7%

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

   if 5.19999999999999954e-4 < b

   1. Initial program 53.2%

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

      1. associate-/l* 53.3%

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

      2. +-commutative 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in a around 0 54.9%

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

      1. *-commutative 54.9%

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

      2. associate-/l* 55.0%

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

   7. Simplified 55.0%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 0.00052:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.3% 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 77.2%

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

   1. +-commutative 77.2%

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

3. Simplified 77.2%

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

   1. *-commutative 77.2%

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

   2. associate-/l* 77.3%

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

6. Applied egg-rr 77.3%

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

Alternative 8: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. associate-/l* 77.2%

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

   2. +-commutative 77.2%

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

3. Simplified 77.2%

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

Alternative 9: 55.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. associate-/l* 77.2%

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

   2. +-commutative 77.2%

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

3. Simplified 77.2%

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

5. Taylor expanded in b around 0 58.7%

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

Alternative 10: 51.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. associate-/l* 77.2%

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

   2. +-commutative 77.2%

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

3. Simplified 77.2%

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

5. Taylor expanded in b around 0 54.9%

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

   1. clear-num 54.9%

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

   2. un-div-inv 54.8%

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

7. Applied egg-rr 54.8%

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

   1. associate-/r/ 54.9%

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

9. Simplified 54.9%

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

10. Final simplification 54.9%

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

Alternative 11: 51.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. associate-/l* 77.2%

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

   2. +-commutative 77.2%

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

3. Simplified 77.2%

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

5. Taylor expanded in b around 0 54.9%

   \[\leadsto r \cdot \color{blue}{\frac{b}{\cos a}} \]
6. Add Preprocessing

Alternative 12: 34.5% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. associate-/l* 77.2%

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

   2. +-commutative 77.2%

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

3. Simplified 77.2%

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

5. Taylor expanded in b around 0 54.9%

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

6. Taylor expanded in a around 0 39.3%

   \[\leadsto r \cdot \color{blue}{b} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024088 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))