rsin A (should all be same)

Percentage Accurate: 76.2% → 99.5%

Time: 17.8s

Alternatives: 15

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: 76.2% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. +-commutative 77.7%

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

3. Simplified 77.7%

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

   1. cos-sum 99.6%

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

   2. *-un-lft-identity 99.6%

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

   3. prod-diff 99.6%

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

6. Applied egg-rr 99.6%

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

   1. *-rgt-identity 99.6%

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

   2. fma-define 99.6%

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

   3. sub-neg 99.6%

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

   4. *-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. fma-undefine 99.6%

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

   7. *-rgt-identity 99.6%

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

   8. distribute-lft-neg-in 99.6%

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

   9. *-rgt-identity 99.6%

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

   10. fma-undefine 99.6%

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

   11. *-commutative 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.3× speedup

Math

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

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 77.7%

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

   1. +-commutative 77.7%

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

3. Simplified 77.7%

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

   1. cos-sum 99.6%

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

   2. cancel-sign-sub-inv 99.6%

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

   3. fma-define 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

   \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(\cos b, \cos a, -\sin b \cdot \sin a\right)} \]
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 77.7%

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

   1. associate-/l* 77.7%

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

   2. remove-double-neg 77.7%

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

   3. remove-double-neg 77.7%

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

   4. +-commutative 77.7%

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

3. Simplified 77.7%

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

   1. cos-sum 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. +-commutative 77.7%

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

3. Simplified 77.7%

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

   1. cos-sum 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 5: 77.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ r \cdot \frac{\sin b}{0.5 \cdot \left(2 \cdot \cos \left(b + a\right)\right) - \sin b \cdot \sin a} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (* r (/ (sin b) (- (* 0.5 (* 2.0 (cos (+ b a)))) (* (sin b) (sin a))))))

C

double code(double r, double a, double b) {
	return r * (sin(b) / ((0.5 * (2.0 * cos((b + 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) / ((0.5d0 * (2.0d0 * cos((b + a)))) - (sin(b) * sin(a))))
end function

Java

public static double code(double r, double a, double b) {
	return r * (Math.sin(b) / ((0.5 * (2.0 * Math.cos((b + a)))) - (Math.sin(b) * Math.sin(a))));
}

Python

def code(r, a, b):
	return r * (math.sin(b) / ((0.5 * (2.0 * math.cos((b + a)))) - (math.sin(b) * math.sin(a))))

Julia

function code(r, a, b)
	return Float64(r * Float64(sin(b) / Float64(Float64(0.5 * Float64(2.0 * cos(Float64(b + a)))) - Float64(sin(b) * sin(a)))))
end

MATLAB

function tmp = code(r, a, b)
	tmp = r * (sin(b) / ((0.5 * (2.0 * cos((b + a)))) - (sin(b) * sin(a))));
end

Wolfram

code[r_, a_, b_] := N[(r * N[(N[Sin[b], $MachinePrecision] / N[(N[(0.5 * N[(2.0 * N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.7%

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

   1. associate-/l* 77.7%

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

   2. remove-double-neg 77.7%

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

   3. remove-double-neg 77.7%

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

   4. +-commutative 77.7%

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

3. Simplified 77.7%

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

   1. cos-sum 99.6%

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

6. Applied egg-rr 99.6%

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

   1. cos-mult 78.7%

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

   2. clear-num 78.7%

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

   3. cos-diff 78.4%

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

   4. add-sqr-sqrt 42.2%

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

   5. sqrt-unprod 78.4%

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

   6. sqr-neg 78.4%

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

   7. sqrt-unprod 36.3%

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

   8. add-sqr-sqrt 78.6%

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

   9. cancel-sign-sub-inv 78.6%

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

   10. cos-sum 78.5%

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

8. Applied egg-rr 78.5%

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

   1. associate-/r/ 78.6%

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

   2. metadata-eval 78.6%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{0.5} \cdot \left(\cos \left(b + a\right) + \cos \left(b + a\right)\right) - \sin b \cdot \sin a} \]

   3. count-2 78.6%

      \[\leadsto r \cdot \frac{\sin b}{0.5 \cdot \color{blue}{\left(2 \cdot \cos \left(b + a\right)\right)} - \sin b \cdot \sin a} \]

10. Simplified 78.6%

    \[\leadsto r \cdot \frac{\sin b}{\color{blue}{0.5 \cdot \left(2 \cdot \cos \left(b + a\right)\right)} - \sin b \cdot \sin a} \]

11. Final simplification 78.6%

    \[\leadsto r \cdot \frac{\sin b}{0.5 \cdot \left(2 \cdot \cos \left(b + a\right)\right) - \sin b \cdot \sin a} \]
12. Add Preprocessing

Alternative 6: 77.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{r \cdot \sin b}{0.5 \cdot \left(2 \cdot \cos \left(b + a\right)\right) - \sin b \cdot \sin a} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (/ (* r (sin b)) (- (* 0.5 (* 2.0 (cos (+ b a)))) (* (sin b) (sin a)))))

C

double code(double r, double a, double b) {
	return (r * sin(b)) / ((0.5 * (2.0 * cos((b + 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)) / ((0.5d0 * (2.0d0 * cos((b + a)))) - (sin(b) * sin(a)))
end function

Java

public static double code(double r, double a, double b) {
	return (r * Math.sin(b)) / ((0.5 * (2.0 * Math.cos((b + a)))) - (Math.sin(b) * Math.sin(a)));
}

Python

def code(r, a, b):
	return (r * math.sin(b)) / ((0.5 * (2.0 * math.cos((b + a)))) - (math.sin(b) * math.sin(a)))

Julia

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / Float64(Float64(0.5 * Float64(2.0 * cos(Float64(b + a)))) - Float64(sin(b) * sin(a))))
end

MATLAB

function tmp = code(r, a, b)
	tmp = (r * sin(b)) / ((0.5 * (2.0 * cos((b + a)))) - (sin(b) * sin(a)));
end

Wolfram

code[r_, a_, b_] := N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[(2.0 * N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.7%

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

   1. +-commutative 77.7%

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

3. Simplified 77.7%

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

   1. cos-sum 99.6%

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

6. Applied egg-rr 99.6%

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

   1. cos-mult 78.7%

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

   2. clear-num 78.7%

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

   3. cos-diff 78.4%

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

   4. add-sqr-sqrt 42.2%

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

   5. sqrt-unprod 78.4%

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

   6. sqr-neg 78.4%

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

   7. sqrt-unprod 36.3%

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

   8. add-sqr-sqrt 78.6%

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

   9. cancel-sign-sub-inv 78.6%

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

   10. cos-sum 78.5%

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

8. Applied egg-rr 78.5%

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

   1. associate-/r/ 78.6%

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

   2. metadata-eval 78.6%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{0.5} \cdot \left(\cos \left(b + a\right) + \cos \left(b + a\right)\right) - \sin b \cdot \sin a} \]

   3. count-2 78.6%

      \[\leadsto r \cdot \frac{\sin b}{0.5 \cdot \color{blue}{\left(2 \cdot \cos \left(b + a\right)\right)} - \sin b \cdot \sin a} \]

10. Simplified 78.6%

    \[\leadsto \frac{r \cdot \sin b}{\color{blue}{0.5 \cdot \left(2 \cdot \cos \left(b + a\right)\right)} - \sin b \cdot \sin a} \]

11. Final simplification 78.6%

    \[\leadsto \frac{r \cdot \sin b}{0.5 \cdot \left(2 \cdot \cos \left(b + a\right)\right) - \sin b \cdot \sin a} \]
12. Add Preprocessing

Alternative 7: 77.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. +-commutative 77.7%

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

3. Simplified 77.7%

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

   1. cos-sum 99.6%

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

   2. cancel-sign-sub-inv 99.6%

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

   3. fma-define 99.6%

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

6. Applied egg-rr 99.6%

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

   1. add-sqr-sqrt 45.4%

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

   2. sqrt-unprod 87.2%

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

   3. sqr-neg 87.2%

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

   4. sqrt-unprod 41.7%

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

   5. add-sqr-sqrt 77.0%

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

   6. sin-mult 78.0%

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

   7. div-sub 78.0%

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

   8. cos-diff 77.3%

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

   9. add-sqr-sqrt 41.9%

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

   10. sqrt-unprod 77.9%

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

   11. sqr-neg 77.9%

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

   12. sqrt-unprod 36.0%

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

   13. add-sqr-sqrt 78.5%

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

   14. cancel-sign-sub-inv 78.5%

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

   15. cos-sum 78.0%

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

8. Applied egg-rr 78.0%

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

   1. +-inverses 78.0%

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

10. Simplified 78.0%

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

11. Final simplification 78.0%

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

Alternative 8: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.9 \cdot 10^{-6} \lor \neg \left(a \leq 165000\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -3.9e-6) (not (<= a 165000.0)))
   (* r (/ (sin b) (cos a)))
   (* r (/ (sin b) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -3.9e-6) || !(a <= 165000.0)) {
		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 <= (-3.9d-6)) .or. (.not. (a <= 165000.0d0))) 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 <= -3.9e-6) || !(a <= 165000.0)) {
		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 <= -3.9e-6) or not (a <= 165000.0):
		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 <= -3.9e-6) || !(a <= 165000.0))
		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 <= -3.9e-6) || ~((a <= 165000.0)))
		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, -3.9e-6], N[Not[LessEqual[a, 165000.0]], $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 < -3.8999999999999999e-6 or 165000 < a

   1. Initial program 53.0%

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

      1. associate-/l* 52.9%

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

      2. remove-double-neg 52.9%

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

      3. remove-double-neg 52.9%

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

      4. +-commutative 52.9%

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

   3. Simplified 52.9%

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

   5. Taylor expanded in b around 0 51.7%

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

   if -3.8999999999999999e-6 < a < 165000

   1. Initial program 98.9%

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

      1. associate-/l* 98.9%

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

      2. remove-double-neg 98.9%

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

      3. remove-double-neg 98.9%

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

      4. +-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 a around 0 98.9%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.9 \cdot 10^{-6} \lor \neg \left(a \leq 165000\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 165000:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -2.7e-6)
   (* r (/ (sin b) (cos a)))
   (if (<= a 165000.0) (* r (/ (sin b) (cos b))) (/ (* r (sin b)) (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -2.7e-6) {
		tmp = r * (sin(b) / cos(a));
	} else if (a <= 165000.0) {
		tmp = r * (sin(b) / cos(b));
	} else {
		tmp = (r * sin(b)) / cos(a);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-2.7d-6)) then
        tmp = r * (sin(b) / cos(a))
    else if (a <= 165000.0d0) then
        tmp = r * (sin(b) / cos(b))
    else
        tmp = (r * sin(b)) / cos(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -2.7e-6) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else if (a <= 165000.0) {
		tmp = r * (Math.sin(b) / Math.cos(b));
	} else {
		tmp = (r * Math.sin(b)) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -2.7e-6:
		tmp = r * (math.sin(b) / math.cos(a))
	elif a <= 165000.0:
		tmp = r * (math.sin(b) / math.cos(b))
	else:
		tmp = (r * math.sin(b)) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -2.7e-6)
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	elseif (a <= 165000.0)
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	else
		tmp = Float64(Float64(r * sin(b)) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -2.7e-6)
		tmp = r * (sin(b) / cos(a));
	elseif (a <= 165000.0)
		tmp = r * (sin(b) / cos(b));
	else
		tmp = (r * sin(b)) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -2.7e-6], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 165000.0], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.69999999999999998e-6

   1. Initial program 59.4%

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

      1. associate-/l* 59.4%

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

      2. remove-double-neg 59.4%

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

      3. remove-double-neg 59.4%

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

      4. +-commutative 59.4%

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

   3. Simplified 59.4%

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

   5. Taylor expanded in b around 0 59.1%

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

   if -2.69999999999999998e-6 < a < 165000

   1. Initial program 98.9%

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

      1. associate-/l* 98.9%

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

      2. remove-double-neg 98.9%

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

      3. remove-double-neg 98.9%

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

      4. +-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 a around 0 98.9%

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

   if 165000 < a

   1. Initial program 45.4%

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

      1. +-commutative 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in b around 0 43.0%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 165000:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;a \leq -3.9 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 165000:\\ \;\;\;\;\frac{t\_0}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= a -3.9e-6)
     (* r (/ (sin b) (cos a)))
     (if (<= a 165000.0) (/ t_0 (cos b)) (/ t_0 (cos a))))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (a <= -3.9e-6) {
		tmp = r * (sin(b) / cos(a));
	} else if (a <= 165000.0) {
		tmp = t_0 / cos(b);
	} else {
		tmp = t_0 / cos(a);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * sin(b)
    if (a <= (-3.9d-6)) then
        tmp = r * (sin(b) / cos(a))
    else if (a <= 165000.0d0) then
        tmp = t_0 / cos(b)
    else
        tmp = t_0 / cos(a)
    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 (a <= -3.9e-6) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else if (a <= 165000.0) {
		tmp = t_0 / Math.cos(b);
	} else {
		tmp = t_0 / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.sin(b)
	tmp = 0
	if a <= -3.9e-6:
		tmp = r * (math.sin(b) / math.cos(a))
	elif a <= 165000.0:
		tmp = t_0 / math.cos(b)
	else:
		tmp = t_0 / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (a <= -3.9e-6)
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	elseif (a <= 165000.0)
		tmp = Float64(t_0 / cos(b));
	else
		tmp = Float64(t_0 / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * sin(b);
	tmp = 0.0;
	if (a <= -3.9e-6)
		tmp = r * (sin(b) / cos(a));
	elseif (a <= 165000.0)
		tmp = t_0 / cos(b);
	else
		tmp = t_0 / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -3.9e-6], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 165000.0], N[(t$95$0 / N[Cos[b], $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[Cos[a], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.8999999999999999e-6

   1. Initial program 59.4%

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

      1. associate-/l* 59.4%

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

      2. remove-double-neg 59.4%

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

      3. remove-double-neg 59.4%

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

      4. +-commutative 59.4%

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

   3. Simplified 59.4%

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

   5. Taylor expanded in b around 0 59.1%

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

   if -3.8999999999999999e-6 < a < 165000

   1. Initial program 98.9%

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

      1. +-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 a around 0 98.9%

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

   if 165000 < a

   1. Initial program 45.4%

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

      1. +-commutative 45.4%

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

   3. Simplified 45.4%

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

   5. Taylor expanded in b around 0 43.0%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.9 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 165000:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.2% 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 77.7%

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

   1. associate-/l* 77.7%

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

   2. remove-double-neg 77.7%

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

   3. remove-double-neg 77.7%

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

   4. +-commutative 77.7%

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

3. Simplified 77.7%

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

5. Final simplification 77.7%

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

Alternative 12: 76.2% 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 77.7%

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

3. Final simplification 77.7%

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

Alternative 13: 54.7% 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 77.7%

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

   1. associate-/l* 77.7%

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

   2. remove-double-neg 77.7%

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

   3. remove-double-neg 77.7%

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

   4. +-commutative 77.7%

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

3. Simplified 77.7%

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

5. Taylor expanded in b around 0 57.6%

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

6. Final simplification 57.6%

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

Alternative 14: 50.7% 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 77.7%

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

   1. +-commutative 77.7%

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

3. Simplified 77.7%

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

   1. cos-sum 99.6%

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

   2. *-un-lft-identity 99.6%

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

   3. prod-diff 99.6%

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

6. Applied egg-rr 99.6%

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

   1. *-rgt-identity 99.6%

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

   2. fma-define 99.6%

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

   3. sub-neg 99.6%

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

   4. *-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. fma-undefine 99.6%

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

   7. *-rgt-identity 99.6%

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

   8. distribute-lft-neg-in 99.6%

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

   9. *-rgt-identity 99.6%

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

   10. fma-undefine 99.6%

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

   11. *-commutative 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in b around 0 54.0%

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

11. Simplified 54.0%

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

12. Final simplification 54.0%

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

Alternative 15: 34.7% 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 77.7%

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

   1. associate-/l* 77.7%

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

   2. remove-double-neg 77.7%

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

   3. remove-double-neg 77.7%

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

   4. +-commutative 77.7%

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

3. Simplified 77.7%

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

5. Taylor expanded in b around 0 54.0%

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

6. Taylor expanded in a around 0 39.6%

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

7. Final simplification 39.6%

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

Reproduce

herbie shell --seed 2024081 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))