rsin B (should all be same)

Percentage Accurate: 76.6% → 99.5%

Time: 14.3s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\sin b\\ \frac{r \cdot \sin b}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \sin a \cdot t_0\right) + \mathsf{fma}\left(t_0, \sin a, \mathsf{expm1}\left(\mathsf{log1p}\left(\sin b \cdot \sin a\right)\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (- (sin b))))
   (/
    (* r (sin b))
    (+
     (fma 1.0 (* (cos a) (cos b)) (* (sin a) t_0))
     (fma t_0 (sin a) (expm1 (log1p (* (sin b) (sin a)))))))))

C

double code(double r, double a, double b) {
	double t_0 = -sin(b);
	return (r * sin(b)) / (fma(1.0, (cos(a) * cos(b)), (sin(a) * t_0)) + fma(t_0, sin(a), expm1(log1p((sin(b) * sin(a))))));
}

Julia

function code(r, a, b)
	t_0 = Float64(-sin(b))
	return Float64(Float64(r * sin(b)) / Float64(fma(1.0, Float64(cos(a) * cos(b)), Float64(sin(a) * t_0)) + fma(t_0, sin(a), expm1(log1p(Float64(sin(b) * sin(a)))))))
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = (-N[Sin[b], $MachinePrecision])}, N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 * N[(N[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[a], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Sin[a], $MachinePrecision] + N[(Exp[N[Log[1 + N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.5%

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

   1. associate-*r/ 75.5%

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

   2. +-commutative 75.5%

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

3. Simplified 75.5%

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

   1. cos-sum 99.4%

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

   2. *-un-lft-identity 99.4%

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

   3. *-un-lft-identity 99.4%

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

   4. prod-diff 99.4%

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

5. Applied egg-rr 99.4%

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

   1. *-rgt-identity 99.4%

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

   2. distribute-lft-neg-in 99.4%

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

   3. *-commutative 99.4%

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

   4. *-commutative 99.4%

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

   5. fma-udef 99.4%

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

   6. *-rgt-identity 99.4%

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

   7. distribute-lft-neg-in 99.4%

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

   8. *-rgt-identity 99.4%

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

   9. fma-udef 99.5%

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

   10. *-commutative 99.5%

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

7. Simplified 99.5%

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

   1. expm1-log1p-u 99.5%

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \sin a \cdot \left(-\sin b\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin a \cdot \sin b\right)\right)}\right)} \]

   2. *-commutative 99.5%

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \sin a \cdot \left(-\sin b\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\sin b \cdot \sin a}\right)\right)\right)} \]

9. Applied egg-rr 99.5%

   \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \sin a \cdot \left(-\sin b\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin b \cdot \sin a\right)\right)}\right)} \]

10. Final simplification 99.5%

    \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \sin a \cdot \left(-\sin b\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \mathsf{expm1}\left(\mathsf{log1p}\left(\sin b \cdot \sin a\right)\right)\right)} \]

Alternative 2: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\sin b\\ \frac{r \cdot \sin b}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \sin a \cdot t_0\right) + \mathsf{fma}\left(t_0, \sin a, \sin b \cdot \sin a\right)} \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (- (sin b))))
   (/
    (* r (sin b))
    (+
     (fma 1.0 (* (cos a) (cos b)) (* (sin a) t_0))
     (fma t_0 (sin a) (* (sin b) (sin a)))))))

C

double code(double r, double a, double b) {
	double t_0 = -sin(b);
	return (r * sin(b)) / (fma(1.0, (cos(a) * cos(b)), (sin(a) * t_0)) + fma(t_0, sin(a), (sin(b) * sin(a))));
}

Julia

function code(r, a, b)
	t_0 = Float64(-sin(b))
	return Float64(Float64(r * sin(b)) / Float64(fma(1.0, Float64(cos(a) * cos(b)), Float64(sin(a) * t_0)) + fma(t_0, sin(a), Float64(sin(b) * sin(a)))))
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = (-N[Sin[b], $MachinePrecision])}, N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 * N[(N[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[a], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Sin[a], $MachinePrecision] + N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 75.5%

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

   1. associate-*r/ 75.5%

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

   2. +-commutative 75.5%

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

3. Simplified 75.5%

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

   1. cos-sum 99.4%

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

   2. *-un-lft-identity 99.4%

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

   3. *-un-lft-identity 99.4%

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

   4. prod-diff 99.4%

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

5. Applied egg-rr 99.4%

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

   1. *-rgt-identity 99.4%

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

   2. distribute-lft-neg-in 99.4%

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

   3. *-commutative 99.4%

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

   4. *-commutative 99.4%

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

   5. fma-udef 99.4%

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

   6. *-rgt-identity 99.4%

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

   7. distribute-lft-neg-in 99.4%

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

   8. *-rgt-identity 99.4%

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

   9. fma-udef 99.5%

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

   10. *-commutative 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 3: 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 75.5%

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

   1. +-commutative 75.5%

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

3. Simplified 75.5%

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

   1. cos-sum 99.4%

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

   2. cancel-sign-sub-inv 99.4%

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

   3. fma-def 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 4: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double r, double a, double b) {
	return (r * sin(b)) * (1.0 / ((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)) * (1.0d0 / ((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)) * (1.0 / ((Math.cos(a) * Math.cos(b)) - (Math.sin(b) * Math.sin(a))));
}

Python

def code(r, a, b):
	return (r * math.sin(b)) * (1.0 / ((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(1.0 / Float64(Float64(cos(a) * cos(b)) - Float64(sin(b) * sin(a)))))
end

MATLAB

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

Wolfram

code[r_, a_, b_] := N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] * N[(1.0 / 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 75.5%

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

   1. *-commutative 75.5%

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

   2. associate-/r/ 75.4%

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

   3. +-commutative 75.4%

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

3. Simplified 75.4%

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

   1. associate-/l* 75.5%

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

   2. *-commutative 75.5%

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

   3. clear-num 75.1%

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

   4. associate-/r/ 75.4%

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

   5. *-commutative 75.4%

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

5. Applied egg-rr 75.4%

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

   1. cos-sum 99.4%

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

7. Applied egg-rr 99.4%

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

8. Final simplification 99.4%

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

Alternative 5: 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 75.5%

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

   1. +-commutative 75.5%

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

3. Simplified 75.5%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 6: 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 75.5%

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

   1. associate-*r/ 75.5%

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

   2. +-commutative 75.5%

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

3. Simplified 75.5%

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

   1. cos-sum 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 7: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0002 \lor \neg \left(b \leq 0.000145\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.0002) (not (<= b 0.000145)))
   (* r (tan b))
   (/ (* r (sin b)) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.0002) || !(b <= 0.000145)) {
		tmp = r * tan(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 ((b <= (-0.0002d0)) .or. (.not. (b <= 0.000145d0))) then
        tmp = r * tan(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 ((b <= -0.0002) || !(b <= 0.000145)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = (r * Math.sin(b)) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.0002) or not (b <= 0.000145):
		tmp = r * math.tan(b)
	else:
		tmp = (r * math.sin(b)) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.0002) || !(b <= 0.000145))
		tmp = Float64(r * tan(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 ((b <= -0.0002) || ~((b <= 0.000145)))
		tmp = r * tan(b);
	else
		tmp = (r * sin(b)) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.0002], N[Not[LessEqual[b, 0.000145]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.0000000000000001e-4 or 1.45e-4 < b

   1. Initial program 51.4%

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

      1. associate-*r/ 51.4%

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

      2. +-commutative 51.4%

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

   3. Simplified 51.4%

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

      1. cos-sum 99.1%

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

      2. *-un-lft-identity 99.1%

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

      3. *-un-lft-identity 99.1%

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

      4. prod-diff 99.1%

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

   5. Applied egg-rr 99.1%

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

      1. *-rgt-identity 99.1%

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

      2. distribute-lft-neg-in 99.1%

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

      3. *-commutative 99.1%

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

      4. *-commutative 99.1%

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

      5. fma-udef 99.1%

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

      6. *-rgt-identity 99.1%

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

      7. distribute-lft-neg-in 99.1%

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

      8. *-rgt-identity 99.1%

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

      9. fma-udef 99.1%

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

      10. *-commutative 99.1%

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

   7. Simplified 99.1%

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

      1. *-commutative 99.1%

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

      2. distribute-rgt-neg-out 99.1%

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

      3. *-commutative 99.1%

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

      4. *-commutative 99.1%

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

      5. prod-diff 99.1%

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

      6. *-un-lft-identity 99.1%

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

      7. *-commutative 99.1%

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

      8. *-commutative 99.1%

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

      9. cos-sum 51.4%

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

      10. div-inv 51.4%

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

      11. *-commutative 51.4%

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

      12. associate-*r* 51.4%

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

   9. Applied egg-rr 51.4%

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

   10. Taylor expanded in a around 0 52.0%

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

       1. expm1-log1p-u 36.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\cos b} \cdot \sin b\right)\right)} \cdot r \]

       2. expm1-udef 36.6%

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

       3. associate-*l/ 36.6%

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

       4. *-un-lft-identity 36.6%

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

       5. quot-tan 36.6%

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

   12. Applied egg-rr 36.6%

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

       1. expm1-def 37.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan b\right)\right)} \cdot r \]

       2. expm1-log1p 52.2%

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

   14. Simplified 52.2%

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

   if -2.0000000000000001e-4 < b < 1.45e-4

   1. Initial program 98.4%

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

      1. associate-*r/ 98.5%

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

      2. +-commutative 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in b around 0 98.5%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0002 \lor \neg \left(b \leq 0.000145\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \end{array} \]

Alternative 8: 76.6% 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 75.5%

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

2. Final simplification 75.5%

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

Alternative 9: 76.6% 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 75.5%

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

   1. *-commutative 75.5%

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

   2. associate-/r/ 75.4%

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

   3. +-commutative 75.4%

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

3. Simplified 75.4%

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

   1. div-inv 75.4%

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

   2. clear-num 75.5%

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

   3. *-commutative 75.5%

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

5. Applied egg-rr 75.5%

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

6. Final simplification 75.5%

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

Alternative 10: 76.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.00014 \lor \neg \left(b \leq 0.000102\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.00014) (not (<= b 0.000102)))
   (* r (tan b))
   (* r (/ b (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.00014) || !(b <= 0.000102)) {
		tmp = r * tan(b);
	} else {
		tmp = r * (b / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-0.00014d0)) .or. (.not. (b <= 0.000102d0))) then
        tmp = r * tan(b)
    else
        tmp = r * (b / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.00014) || !(b <= 0.000102)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = r * (b / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.00014) or not (b <= 0.000102):
		tmp = r * math.tan(b)
	else:
		tmp = r * (b / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.00014) || !(b <= 0.000102))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(r * Float64(b / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.00014) || ~((b <= 0.000102)))
		tmp = r * tan(b);
	else
		tmp = r * (b / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.00014], N[Not[LessEqual[b, 0.000102]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.3999999999999999e-4 or 1.01999999999999999e-4 < b

   1. Initial program 51.4%

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

      1. associate-*r/ 51.4%

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

      2. +-commutative 51.4%

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

   3. Simplified 51.4%

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

      1. cos-sum 99.1%

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

      2. *-un-lft-identity 99.1%

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

      3. *-un-lft-identity 99.1%

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

      4. prod-diff 99.1%

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

   5. Applied egg-rr 99.1%

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

      1. *-rgt-identity 99.1%

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

      2. distribute-lft-neg-in 99.1%

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

      3. *-commutative 99.1%

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

      4. *-commutative 99.1%

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

      5. fma-udef 99.1%

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

      6. *-rgt-identity 99.1%

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

      7. distribute-lft-neg-in 99.1%

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

      8. *-rgt-identity 99.1%

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

      9. fma-udef 99.1%

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

      10. *-commutative 99.1%

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

   7. Simplified 99.1%

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

      1. *-commutative 99.1%

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

      2. distribute-rgt-neg-out 99.1%

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

      3. *-commutative 99.1%

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

      4. *-commutative 99.1%

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

      5. prod-diff 99.1%

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

      6. *-un-lft-identity 99.1%

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

      7. *-commutative 99.1%

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

      8. *-commutative 99.1%

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

      9. cos-sum 51.4%

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

      10. div-inv 51.4%

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

      11. *-commutative 51.4%

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

      12. associate-*r* 51.4%

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

   9. Applied egg-rr 51.4%

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

   10. Taylor expanded in a around 0 52.0%

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

       1. expm1-log1p-u 36.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\cos b} \cdot \sin b\right)\right)} \cdot r \]

       2. expm1-udef 36.6%

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

       3. associate-*l/ 36.6%

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

       4. *-un-lft-identity 36.6%

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

       5. quot-tan 36.6%

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

   12. Applied egg-rr 36.6%

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

       1. expm1-def 37.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan b\right)\right)} \cdot r \]

       2. expm1-log1p 52.2%

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

   14. Simplified 52.2%

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

   if -1.3999999999999999e-4 < b < 1.01999999999999999e-4

   1. Initial program 98.4%

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

      1. +-commutative 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in b around 0 98.4%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.00014 \lor \neg \left(b \leq 0.000102\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 11: 76.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.00026 \lor \neg \left(b \leq 0.000112\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.00026) (not (<= b 0.000112)))
   (* r (tan b))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.00026) || !(b <= 0.000112)) {
		tmp = r * tan(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 <= (-0.00026d0)) .or. (.not. (b <= 0.000112d0))) then
        tmp = r * tan(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 <= -0.00026) || !(b <= 0.000112)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.00026) or not (b <= 0.000112):
		tmp = r * math.tan(b)
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.00026) || !(b <= 0.000112))
		tmp = Float64(r * tan(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 <= -0.00026) || ~((b <= 0.000112)))
		tmp = r * tan(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.00026], N[Not[LessEqual[b, 0.000112]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.59999999999999977e-4 or 1.11999999999999998e-4 < b

   1. Initial program 51.4%

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

      1. associate-*r/ 51.4%

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

      2. +-commutative 51.4%

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

   3. Simplified 51.4%

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

      1. cos-sum 99.1%

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

      2. *-un-lft-identity 99.1%

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

      3. *-un-lft-identity 99.1%

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

      4. prod-diff 99.1%

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

   5. Applied egg-rr 99.1%

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

      1. *-rgt-identity 99.1%

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

      2. distribute-lft-neg-in 99.1%

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

      3. *-commutative 99.1%

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

      4. *-commutative 99.1%

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

      5. fma-udef 99.1%

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

      6. *-rgt-identity 99.1%

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

      7. distribute-lft-neg-in 99.1%

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

      8. *-rgt-identity 99.1%

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

      9. fma-udef 99.1%

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

      10. *-commutative 99.1%

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

   7. Simplified 99.1%

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

      1. *-commutative 99.1%

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

      2. distribute-rgt-neg-out 99.1%

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

      3. *-commutative 99.1%

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

      4. *-commutative 99.1%

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

      5. prod-diff 99.1%

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

      6. *-un-lft-identity 99.1%

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

      7. *-commutative 99.1%

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

      8. *-commutative 99.1%

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

      9. cos-sum 51.4%

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

      10. div-inv 51.4%

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

      11. *-commutative 51.4%

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

      12. associate-*r* 51.4%

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

   9. Applied egg-rr 51.4%

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

   10. Taylor expanded in a around 0 52.0%

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

       1. expm1-log1p-u 36.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\cos b} \cdot \sin b\right)\right)} \cdot r \]

       2. expm1-udef 36.6%

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

       3. associate-*l/ 36.6%

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

       4. *-un-lft-identity 36.6%

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

       5. quot-tan 36.6%

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

   12. Applied egg-rr 36.6%

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

       1. expm1-def 37.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan b\right)\right)} \cdot r \]

       2. expm1-log1p 52.2%

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

   14. Simplified 52.2%

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

   if -2.59999999999999977e-4 < b < 1.11999999999999998e-4

   1. Initial program 98.4%

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

      1. +-commutative 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in b around 0 98.5%

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

      1. *-commutative 98.5%

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

      2. associate-/l* 98.3%

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

      3. associate-/r/ 98.5%

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

   6. Simplified 98.5%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.00026 \lor \neg \left(b \leq 0.000112\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 12: 76.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.00014 \lor \neg \left(b \leq 0.000102\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.00014) (not (<= b 0.000102)))
   (* r (tan b))
   (/ (* r b) (cos a))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.00014) || !(b <= 0.000102)) {
		tmp = r * tan(b);
	} else {
		tmp = (r * b) / cos(a);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-0.00014d0)) .or. (.not. (b <= 0.000102d0))) then
        tmp = r * tan(b)
    else
        tmp = (r * b) / cos(a)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.00014) || !(b <= 0.000102)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = (r * b) / Math.cos(a);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.00014) or not (b <= 0.000102):
		tmp = r * math.tan(b)
	else:
		tmp = (r * b) / math.cos(a)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.00014) || !(b <= 0.000102))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(Float64(r * b) / cos(a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.00014) || ~((b <= 0.000102)))
		tmp = r * tan(b);
	else
		tmp = (r * b) / cos(a);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.00014], N[Not[LessEqual[b, 0.000102]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(N[(r * b), $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.3999999999999999e-4 or 1.01999999999999999e-4 < b

   1. Initial program 51.4%

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

      1. associate-*r/ 51.4%

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

      2. +-commutative 51.4%

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

   3. Simplified 51.4%

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

      1. cos-sum 99.1%

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

      2. *-un-lft-identity 99.1%

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

      3. *-un-lft-identity 99.1%

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

      4. prod-diff 99.1%

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

   5. Applied egg-rr 99.1%

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

      1. *-rgt-identity 99.1%

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

      2. distribute-lft-neg-in 99.1%

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

      3. *-commutative 99.1%

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

      4. *-commutative 99.1%

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

      5. fma-udef 99.1%

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

      6. *-rgt-identity 99.1%

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

      7. distribute-lft-neg-in 99.1%

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

      8. *-rgt-identity 99.1%

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

      9. fma-udef 99.1%

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

      10. *-commutative 99.1%

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

   7. Simplified 99.1%

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

      1. *-commutative 99.1%

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

      2. distribute-rgt-neg-out 99.1%

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

      3. *-commutative 99.1%

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

      4. *-commutative 99.1%

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

      5. prod-diff 99.1%

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

      6. *-un-lft-identity 99.1%

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

      7. *-commutative 99.1%

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

      8. *-commutative 99.1%

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

      9. cos-sum 51.4%

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

      10. div-inv 51.4%

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

      11. *-commutative 51.4%

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

      12. associate-*r* 51.4%

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

   9. Applied egg-rr 51.4%

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

   10. Taylor expanded in a around 0 52.0%

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

       1. expm1-log1p-u 36.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\cos b} \cdot \sin b\right)\right)} \cdot r \]

       2. expm1-udef 36.6%

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

       3. associate-*l/ 36.6%

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

       4. *-un-lft-identity 36.6%

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

       5. quot-tan 36.6%

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

   12. Applied egg-rr 36.6%

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

       1. expm1-def 37.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan b\right)\right)} \cdot r \]

       2. expm1-log1p 52.2%

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

   14. Simplified 52.2%

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

   if -1.3999999999999999e-4 < b < 1.01999999999999999e-4

   1. Initial program 98.4%

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

      1. +-commutative 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in b around 0 98.5%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.00014 \lor \neg \left(b \leq 0.000102\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \]

Alternative 13: 60.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double r, double a, double b) {
	return r * tan(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 * tan(b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.5%

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

   1. associate-*r/ 75.5%

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

   2. +-commutative 75.5%

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

3. Simplified 75.5%

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

   1. cos-sum 99.4%

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

   2. *-un-lft-identity 99.4%

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

   3. *-un-lft-identity 99.4%

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

   4. prod-diff 99.4%

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

5. Applied egg-rr 99.4%

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

   1. *-rgt-identity 99.4%

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

   2. distribute-lft-neg-in 99.4%

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

   3. *-commutative 99.4%

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

   4. *-commutative 99.4%

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

   5. fma-udef 99.4%

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

   6. *-rgt-identity 99.4%

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

   7. distribute-lft-neg-in 99.4%

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

   8. *-rgt-identity 99.4%

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

   9. fma-udef 99.5%

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

   10. *-commutative 99.5%

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

7. Simplified 99.5%

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

   1. *-commutative 99.5%

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

   2. distribute-rgt-neg-out 99.5%

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

   3. *-commutative 99.5%

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

   4. *-commutative 99.5%

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

   5. prod-diff 99.4%

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

   6. *-un-lft-identity 99.4%

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

   7. *-commutative 99.4%

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

   8. *-commutative 99.4%

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

   9. cos-sum 75.5%

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

   10. div-inv 75.4%

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

   11. *-commutative 75.4%

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

   12. associate-*r* 75.4%

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

9. Applied egg-rr 75.4%

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

10. Taylor expanded in a around 0 61.6%

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

    1. expm1-log1p-u 54.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\cos b} \cdot \sin b\right)\right)} \cdot r \]

    2. expm1-udef 33.0%

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

    3. associate-*l/ 33.0%

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

    4. *-un-lft-identity 33.0%

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

    5. quot-tan 33.0%

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

12. Applied egg-rr 33.0%

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

    1. expm1-def 54.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan b\right)\right)} \cdot r \]

    2. expm1-log1p 61.7%

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

14. Simplified 61.7%

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

15. Final simplification 61.7%

    \[\leadsto r \cdot \tan b \]

Alternative 14: 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 75.5%

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

   1. +-commutative 75.5%

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

3. Simplified 75.5%

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

4. Taylor expanded in b around 0 52.3%

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

5. Taylor expanded in a around 0 37.9%

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

6. Final simplification 37.9%

   \[\leadsto r \cdot b \]

Reproduce

herbie shell --seed 2023257 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))