rsin B (should all be same)

Percentage Accurate: 76.8% → 99.5%

Time: 14.0s

Alternatives: 15

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: 76.8% 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, 0.4× speedup

Math

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

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

Derivation

1. Initial program 75.9%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in r around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-sin.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 3: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.0095:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \mathbf{elif}\;b \leq 0.0195:\\ \;\;\;\;\frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b, -0.16666666666666666 \cdot \left(b \cdot b\right), b\right)\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.0095)
   (/ (* r (sin b)) (cos b))
   (if (<= b 0.0195)
     (* (/ r (cos (+ b a))) (fma b (* -0.16666666666666666 (* b b)) b))
     (* r (/ (sin b) (cos b))))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.0095) {
		tmp = (r * sin(b)) / cos(b);
	} else if (b <= 0.0195) {
		tmp = (r / cos((b + a))) * fma(b, (-0.16666666666666666 * (b * b)), b);
	} else {
		tmp = r * (sin(b) / cos(b));
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.0095)
		tmp = Float64(Float64(r * sin(b)) / cos(b));
	elseif (b <= 0.0195)
		tmp = Float64(Float64(r / cos(Float64(b + a))) * fma(b, Float64(-0.16666666666666666 * Float64(b * b)), b));
	else
		tmp = Float64(r * Float64(sin(b) / cos(b)));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.0095], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.0195], N[(N[(r / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b * N[(-0.16666666666666666 * N[(b * b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.00949999999999999976

   1. Initial program 41.1%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-cos.f64 42.9

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

   5. Applied rewrites 42.9%

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

   if -0.00949999999999999976 < b < 0.0195

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 98.6

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in b around 0

      \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\left(b \cdot \left(1 + \frac{-1}{6} \cdot {b}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \left(b \cdot \color{blue}{\left(\frac{-1}{6} \cdot {b}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\left(b \cdot \left(\frac{-1}{6} \cdot {b}^{2}\right) + b \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \left(b \cdot \left(\frac{-1}{6} \cdot {b}^{2}\right) + \color{blue}{b}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\mathsf{fma}\left(b, \frac{-1}{6} \cdot {b}^{2}, b\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b, \color{blue}{\frac{-1}{6} \cdot {b}^{2}}, b\right) \]

      6. unpow2 N/A

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

      7. lower-*.f64 98.6

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b, -0.16666666666666666 \cdot \color{blue}{\left(b \cdot b\right)}, b\right) \]

   7. Applied rewrites 98.6%

      \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\mathsf{fma}\left(b, -0.16666666666666666 \cdot \left(b \cdot b\right), b\right)} \]

   if 0.0195 < b

   1. Initial program 58.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-cos.f64 58.8

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

   5. Applied rewrites 58.8%

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

Alternative 4: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r \cdot \sin b}{\cos b}\\ \mathbf{if}\;b \leq -0.0095:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.0195:\\ \;\;\;\;\frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b, -0.16666666666666666 \cdot \left(b \cdot b\right), b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (* r (sin b)) (cos b))))
   (if (<= b -0.0095)
     t_0
     (if (<= b 0.0195)
       (* (/ r (cos (+ b a))) (fma b (* -0.16666666666666666 (* b b)) b))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r * sin(b)) / cos(b);
	double tmp;
	if (b <= -0.0095) {
		tmp = t_0;
	} else if (b <= 0.0195) {
		tmp = (r / cos((b + a))) * fma(b, (-0.16666666666666666 * (b * b)), b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r * sin(b)) / cos(b))
	tmp = 0.0
	if (b <= -0.0095)
		tmp = t_0;
	elseif (b <= 0.0195)
		tmp = Float64(Float64(r / cos(Float64(b + a))) * fma(b, Float64(-0.16666666666666666 * Float64(b * b)), b));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.0095], t$95$0, If[LessEqual[b, 0.0195], N[(N[(r / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b * N[(-0.16666666666666666 * N[(b * b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.00949999999999999976 or 0.0195 < b

   1. Initial program 48.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-cos.f64 49.9

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

   5. Applied rewrites 49.9%

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

   if -0.00949999999999999976 < b < 0.0195

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 98.6

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in b around 0

      \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\left(b \cdot \left(1 + \frac{-1}{6} \cdot {b}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \left(b \cdot \color{blue}{\left(\frac{-1}{6} \cdot {b}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\left(b \cdot \left(\frac{-1}{6} \cdot {b}^{2}\right) + b \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \left(b \cdot \left(\frac{-1}{6} \cdot {b}^{2}\right) + \color{blue}{b}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\mathsf{fma}\left(b, \frac{-1}{6} \cdot {b}^{2}, b\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b, \color{blue}{\frac{-1}{6} \cdot {b}^{2}}, b\right) \]

      6. unpow2 N/A

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

      7. lower-*.f64 98.6

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b, -0.16666666666666666 \cdot \color{blue}{\left(b \cdot b\right)}, b\right) \]

   7. Applied rewrites 98.6%

      \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\mathsf{fma}\left(b, -0.16666666666666666 \cdot \left(b \cdot b\right), b\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 75.9

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 75.9

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

4. Applied rewrites 75.9%

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

Alternative 6: 76.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*l/ N/A

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

   9. *-lft-identity N/A

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

   10. lower-/.f64 75.9

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 75.9

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

4. Applied rewrites 75.9%

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

5. Final simplification 75.9%

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

Alternative 7: 76.8% 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 75.9%

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

3. Final simplification 75.9%

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

Alternative 8: 75.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a + \frac{-1}{\tan b}\\ \mathbf{if}\;b \leq -0.011:\\ \;\;\;\;r \cdot \frac{-1}{t\_0}\\ \mathbf{elif}\;b \leq 0.023:\\ \;\;\;\;\frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b, -0.16666666666666666 \cdot \left(b \cdot b\right), b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{r}}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (+ a (/ -1.0 (tan b)))))
   (if (<= b -0.011)
     (* r (/ -1.0 t_0))
     (if (<= b 0.023)
       (* (/ r (cos (+ b a))) (fma b (* -0.16666666666666666 (* b b)) b))
       (/ -1.0 (/ t_0 r))))))

C

double code(double r, double a, double b) {
	double t_0 = a + (-1.0 / tan(b));
	double tmp;
	if (b <= -0.011) {
		tmp = r * (-1.0 / t_0);
	} else if (b <= 0.023) {
		tmp = (r / cos((b + a))) * fma(b, (-0.16666666666666666 * (b * b)), b);
	} else {
		tmp = -1.0 / (t_0 / r);
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(a + Float64(-1.0 / tan(b)))
	tmp = 0.0
	if (b <= -0.011)
		tmp = Float64(r * Float64(-1.0 / t_0));
	elseif (b <= 0.023)
		tmp = Float64(Float64(r / cos(Float64(b + a))) * fma(b, Float64(-0.16666666666666666 * Float64(b * b)), b));
	else
		tmp = Float64(-1.0 / Float64(t_0 / r));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(a + N[(-1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.011], N[(r * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.023], N[(N[(r / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b * N[(-0.16666666666666666 * N[(b * b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(t$95$0 / r), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.010999999999999999

   1. Initial program 41.1%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. unsub-neg N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. cos-sum N/A

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

      15. lift-+.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. frac-2neg N/A

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

   6. Applied rewrites 41.0%

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

   7. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 38.2

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

   9. Applied rewrites 38.2%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-neg.f64 N/A

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

       4. distribute-frac-neg2 N/A

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

       5. distribute-frac-neg N/A

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

       6. associate-/r/ N/A

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

       7. lower-*.f64 N/A

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

   11. Applied rewrites 38.4%

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

   if -0.010999999999999999 < b < 0.023

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 98.6

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in b around 0

      \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\left(b \cdot \left(1 + \frac{-1}{6} \cdot {b}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \left(b \cdot \color{blue}{\left(\frac{-1}{6} \cdot {b}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\left(b \cdot \left(\frac{-1}{6} \cdot {b}^{2}\right) + b \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \left(b \cdot \left(\frac{-1}{6} \cdot {b}^{2}\right) + \color{blue}{b}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\mathsf{fma}\left(b, \frac{-1}{6} \cdot {b}^{2}, b\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b, \color{blue}{\frac{-1}{6} \cdot {b}^{2}}, b\right) \]

      6. unpow2 N/A

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

      7. lower-*.f64 98.6

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b, -0.16666666666666666 \cdot \color{blue}{\left(b \cdot b\right)}, b\right) \]

   7. Applied rewrites 98.6%

      \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\mathsf{fma}\left(b, -0.16666666666666666 \cdot \left(b \cdot b\right), b\right)} \]

   if 0.023 < b

   1. Initial program 58.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 99.3

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

   4. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. unsub-neg N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. cos-sum N/A

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

      15. lift-+.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. frac-2neg N/A

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

   6. Applied rewrites 57.9%

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

   7. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 54.8

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

   9. Applied rewrites 54.8%

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

       1. lift-/.f64 N/A

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

       2. frac-2neg N/A

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

       3. metadata-eval N/A

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

       4. lower-/.f64 N/A

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

       5. lift-/.f64 N/A

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

       6. lift-neg.f64 N/A

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

       7. distribute-frac-neg2 N/A

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

       8. remove-double-neg N/A

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

       9. lower-/.f64 54.8

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

   11. Applied rewrites 54.9%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.011:\\ \;\;\;\;r \cdot \frac{-1}{a + \frac{-1}{\tan b}}\\ \mathbf{elif}\;b \leq 0.023:\\ \;\;\;\;\frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b, -0.16666666666666666 \cdot \left(b \cdot b\right), b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{a + \frac{-1}{\tan b}}{r}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a + \frac{-1}{\tan b}\\ \mathbf{if}\;b \leq -0.011:\\ \;\;\;\;r \cdot \frac{-1}{t\_0}\\ \mathbf{elif}\;b \leq 0.023:\\ \;\;\;\;\frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b, -0.16666666666666666 \cdot \left(b \cdot b\right), b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-r}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (+ a (/ -1.0 (tan b)))))
   (if (<= b -0.011)
     (* r (/ -1.0 t_0))
     (if (<= b 0.023)
       (* (/ r (cos (+ b a))) (fma b (* -0.16666666666666666 (* b b)) b))
       (/ (- r) t_0)))))

C

double code(double r, double a, double b) {
	double t_0 = a + (-1.0 / tan(b));
	double tmp;
	if (b <= -0.011) {
		tmp = r * (-1.0 / t_0);
	} else if (b <= 0.023) {
		tmp = (r / cos((b + a))) * fma(b, (-0.16666666666666666 * (b * b)), b);
	} else {
		tmp = -r / t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(a + Float64(-1.0 / tan(b)))
	tmp = 0.0
	if (b <= -0.011)
		tmp = Float64(r * Float64(-1.0 / t_0));
	elseif (b <= 0.023)
		tmp = Float64(Float64(r / cos(Float64(b + a))) * fma(b, Float64(-0.16666666666666666 * Float64(b * b)), b));
	else
		tmp = Float64(Float64(-r) / t_0);
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(a + N[(-1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.011], N[(r * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.023], N[(N[(r / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b * N[(-0.16666666666666666 * N[(b * b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], N[((-r) / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.010999999999999999

   1. Initial program 41.1%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. unsub-neg N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. cos-sum N/A

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

      15. lift-+.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. frac-2neg N/A

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

   6. Applied rewrites 41.0%

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

   7. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 38.2

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

   9. Applied rewrites 38.2%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-neg.f64 N/A

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

       4. distribute-frac-neg2 N/A

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

       5. distribute-frac-neg N/A

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

       6. associate-/r/ N/A

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

       7. lower-*.f64 N/A

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

   11. Applied rewrites 38.4%

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

   if -0.010999999999999999 < b < 0.023

   1. Initial program 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. lower-/.f64 98.6

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in b around 0

      \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\left(b \cdot \left(1 + \frac{-1}{6} \cdot {b}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \left(b \cdot \color{blue}{\left(\frac{-1}{6} \cdot {b}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\left(b \cdot \left(\frac{-1}{6} \cdot {b}^{2}\right) + b \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \left(b \cdot \left(\frac{-1}{6} \cdot {b}^{2}\right) + \color{blue}{b}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\mathsf{fma}\left(b, \frac{-1}{6} \cdot {b}^{2}, b\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b, \color{blue}{\frac{-1}{6} \cdot {b}^{2}}, b\right) \]

      6. unpow2 N/A

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

      7. lower-*.f64 98.6

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b, -0.16666666666666666 \cdot \color{blue}{\left(b \cdot b\right)}, b\right) \]

   7. Applied rewrites 98.6%

      \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\mathsf{fma}\left(b, -0.16666666666666666 \cdot \left(b \cdot b\right), b\right)} \]

   if 0.023 < b

   1. Initial program 58.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 99.3

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

   4. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. unsub-neg N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. cos-sum N/A

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

      15. lift-+.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. frac-2neg N/A

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

   6. Applied rewrites 57.9%

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

   7. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 54.8

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

   9. Applied rewrites 54.8%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. clear-num N/A

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

       4. lower-/.f64 54.9

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

   11. Applied rewrites 54.9%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.011:\\ \;\;\;\;r \cdot \frac{-1}{a + \frac{-1}{\tan b}}\\ \mathbf{elif}\;b \leq 0.023:\\ \;\;\;\;\frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b, -0.16666666666666666 \cdot \left(b \cdot b\right), b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-r}{a + \frac{-1}{\tan b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a + \frac{-1}{\tan b}\\ \mathbf{if}\;b \leq -0.011:\\ \;\;\;\;r \cdot \frac{-1}{t\_0}\\ \mathbf{elif}\;b \leq 0.023:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(b, b \cdot \left(b \cdot -0.16666666666666666\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-r}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (+ a (/ -1.0 (tan b)))))
   (if (<= b -0.011)
     (* r (/ -1.0 t_0))
     (if (<= b 0.023)
       (* r (/ (fma b (* b (* b -0.16666666666666666)) b) (cos (+ b a))))
       (/ (- r) t_0)))))

C

double code(double r, double a, double b) {
	double t_0 = a + (-1.0 / tan(b));
	double tmp;
	if (b <= -0.011) {
		tmp = r * (-1.0 / t_0);
	} else if (b <= 0.023) {
		tmp = r * (fma(b, (b * (b * -0.16666666666666666)), b) / cos((b + a)));
	} else {
		tmp = -r / t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(a + Float64(-1.0 / tan(b)))
	tmp = 0.0
	if (b <= -0.011)
		tmp = Float64(r * Float64(-1.0 / t_0));
	elseif (b <= 0.023)
		tmp = Float64(r * Float64(fma(b, Float64(b * Float64(b * -0.16666666666666666)), b) / cos(Float64(b + a))));
	else
		tmp = Float64(Float64(-r) / t_0);
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(a + N[(-1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.011], N[(r * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.023], N[(r * N[(N[(b * N[(b * N[(b * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-r) / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.010999999999999999

   1. Initial program 41.1%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. unsub-neg N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. cos-sum N/A

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

      15. lift-+.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. frac-2neg N/A

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

   6. Applied rewrites 41.0%

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

   7. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 38.2

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

   9. Applied rewrites 38.2%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-neg.f64 N/A

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

       4. distribute-frac-neg2 N/A

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

       5. distribute-frac-neg N/A

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

       6. associate-/r/ N/A

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

       7. lower-*.f64 N/A

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

   11. Applied rewrites 38.4%

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

   if -0.010999999999999999 < b < 0.023

   1. Initial program 98.6%

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

   3. Taylor expanded in b around 0

      \[\leadsto r \cdot \frac{\color{blue}{b \cdot \left(1 + \frac{-1}{6} \cdot {b}^{2}\right)}}{\cos \left(a + b\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto r \cdot \frac{b \cdot \color{blue}{\left(\frac{-1}{6} \cdot {b}^{2} + 1\right)}}{\cos \left(a + b\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto r \cdot \frac{\color{blue}{b \cdot \left(\frac{-1}{6} \cdot {b}^{2}\right) + b \cdot 1}}{\cos \left(a + b\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto r \cdot \frac{b \cdot \left(\frac{-1}{6} \cdot {b}^{2}\right) + \color{blue}{b}}{\cos \left(a + b\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left(b, \frac{-1}{6} \cdot {b}^{2}, b\right)}}{\cos \left(a + b\right)} \]

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 98.6

          \[\leadsto r \cdot \frac{\mathsf{fma}\left(b, b \cdot \color{blue}{\left(b \cdot -0.16666666666666666\right)}, b\right)}{\cos \left(a + b\right)} \]

   5. Applied rewrites 98.6%

      \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left(b, b \cdot \left(b \cdot -0.16666666666666666\right), b\right)}}{\cos \left(a + b\right)} \]

   if 0.023 < b

   1. Initial program 58.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 99.3

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

   4. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. unsub-neg N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. cos-sum N/A

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

      15. lift-+.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. frac-2neg N/A

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

   6. Applied rewrites 57.9%

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

   7. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 54.8

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

   9. Applied rewrites 54.8%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. clear-num N/A

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

       4. lower-/.f64 54.9

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

   11. Applied rewrites 54.9%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.011:\\ \;\;\;\;r \cdot \frac{-1}{a + \frac{-1}{\tan b}}\\ \mathbf{elif}\;b \leq 0.023:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(b, b \cdot \left(b \cdot -0.16666666666666666\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-r}{a + \frac{-1}{\tan b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a + \frac{-1}{\tan b}\\ \mathbf{if}\;b \leq -0.0016:\\ \;\;\;\;r \cdot \frac{-1}{t\_0}\\ \mathbf{elif}\;b \leq 0.0061:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-r}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (+ a (/ -1.0 (tan b)))))
   (if (<= b -0.0016)
     (* r (/ -1.0 t_0))
     (if (<= b 0.0061) (* b (/ r (cos a))) (/ (- r) t_0)))))

C

double code(double r, double a, double b) {
	double t_0 = a + (-1.0 / tan(b));
	double tmp;
	if (b <= -0.0016) {
		tmp = r * (-1.0 / t_0);
	} else if (b <= 0.0061) {
		tmp = b * (r / cos(a));
	} else {
		tmp = -r / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a + ((-1.0d0) / tan(b))
    if (b <= (-0.0016d0)) then
        tmp = r * ((-1.0d0) / t_0)
    else if (b <= 0.0061d0) then
        tmp = b * (r / cos(a))
    else
        tmp = -r / t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = a + (-1.0 / Math.tan(b));
	double tmp;
	if (b <= -0.0016) {
		tmp = r * (-1.0 / t_0);
	} else if (b <= 0.0061) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = -r / t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = a + (-1.0 / math.tan(b))
	tmp = 0
	if b <= -0.0016:
		tmp = r * (-1.0 / t_0)
	elif b <= 0.0061:
		tmp = b * (r / math.cos(a))
	else:
		tmp = -r / t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(a + Float64(-1.0 / tan(b)))
	tmp = 0.0
	if (b <= -0.0016)
		tmp = Float64(r * Float64(-1.0 / t_0));
	elseif (b <= 0.0061)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = Float64(Float64(-r) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = a + (-1.0 / tan(b));
	tmp = 0.0;
	if (b <= -0.0016)
		tmp = r * (-1.0 / t_0);
	elseif (b <= 0.0061)
		tmp = b * (r / cos(a));
	else
		tmp = -r / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(a + N[(-1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.0016], N[(r * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.0061], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-r) / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.00160000000000000008

   1. Initial program 41.1%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. unsub-neg N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. cos-sum N/A

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

      15. lift-+.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. frac-2neg N/A

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

   6. Applied rewrites 41.0%

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

   7. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 38.2

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

   9. Applied rewrites 38.2%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-neg.f64 N/A

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

       4. distribute-frac-neg2 N/A

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

       5. distribute-frac-neg N/A

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

       6. associate-/r/ N/A

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

       7. lower-*.f64 N/A

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

   11. Applied rewrites 38.4%

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

   if -0.00160000000000000008 < b < 0.00610000000000000039

   1. Initial program 98.6%

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

   3. Taylor expanded in b around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cos.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if 0.00610000000000000039 < b

   1. Initial program 58.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 99.3

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

   4. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. unsub-neg N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. cos-sum N/A

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

      15. lift-+.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. frac-2neg N/A

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

   6. Applied rewrites 57.9%

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

   7. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 54.8

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

   9. Applied rewrites 54.8%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. clear-num N/A

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

       4. lower-/.f64 54.9

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

   11. Applied rewrites 54.9%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0016:\\ \;\;\;\;r \cdot \frac{-1}{a + \frac{-1}{\tan b}}\\ \mathbf{elif}\;b \leq 0.0061:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-r}{a + \frac{-1}{\tan b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-r}{a + \frac{-1}{\tan b}}\\ \mathbf{if}\;b \leq -0.0016:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.0061:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (- r) (+ a (/ -1.0 (tan b))))))
   (if (<= b -0.0016) t_0 (if (<= b 0.0061) (* b (/ r (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = -r / (a + (-1.0 / tan(b)));
	double tmp;
	if (b <= -0.0016) {
		tmp = t_0;
	} else if (b <= 0.0061) {
		tmp = b * (r / cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -r / (a + ((-1.0d0) / tan(b)))
    if (b <= (-0.0016d0)) then
        tmp = t_0
    else if (b <= 0.0061d0) then
        tmp = b * (r / cos(a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = -r / (a + (-1.0 / Math.tan(b)));
	double tmp;
	if (b <= -0.0016) {
		tmp = t_0;
	} else if (b <= 0.0061) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = -r / (a + (-1.0 / math.tan(b)))
	tmp = 0
	if b <= -0.0016:
		tmp = t_0
	elif b <= 0.0061:
		tmp = b * (r / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(-r) / Float64(a + Float64(-1.0 / tan(b))))
	tmp = 0.0
	if (b <= -0.0016)
		tmp = t_0;
	elseif (b <= 0.0061)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = -r / (a + (-1.0 / tan(b)));
	tmp = 0.0;
	if (b <= -0.0016)
		tmp = t_0;
	elseif (b <= 0.0061)
		tmp = b * (r / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[((-r) / N[(a + N[(-1.0 / N[Tan[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.0016], t$95$0, If[LessEqual[b, 0.0061], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.00160000000000000008 or 0.00610000000000000039 < b

   1. Initial program 48.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-*.f64 N/A

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. unsub-neg N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. cos-sum N/A

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

      15. lift-+.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. frac-2neg N/A

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

   6. Applied rewrites 48.4%

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

   7. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 45.5

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

   9. Applied rewrites 45.5%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. clear-num N/A

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

       4. lower-/.f64 45.6

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

   11. Applied rewrites 45.6%

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

   if -0.00160000000000000008 < b < 0.00610000000000000039

   1. Initial program 98.6%

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

   3. Taylor expanded in b around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cos.f64 98.4

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

   5. Applied rewrites 98.4%

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

Alternative 13: 53.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9000:\\ \;\;\;\;r \cdot \frac{\sin b}{1}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -9000.0) (* r (/ (sin b) 1.0)) (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -9000.0) {
		tmp = r * (sin(b) / 1.0);
	} 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 <= (-9000.0d0)) then
        tmp = r * (sin(b) / 1.0d0)
    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 <= -9000.0) {
		tmp = r * (Math.sin(b) / 1.0);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -9000.0:
		tmp = r * (math.sin(b) / 1.0)
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -9000.0)
		tmp = Float64(r * Float64(sin(b) / 1.0));
	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 <= -9000.0)
		tmp = r * (sin(b) / 1.0);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -9000.0], N[(r * N[(N[Sin[b], $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9e3

   1. Initial program 41.1%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. remove-double-div N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 99.2

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in b around 0

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

      1. lower-cos.f64 11.1

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

   9. Applied rewrites 11.1%

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

   10. Taylor expanded in a around 0

       \[\leadsto r \cdot \frac{\sin b}{1} \]
   11. Step-by-step derivation

   12. Applied rewrites 10.6%

       \[\leadsto r \cdot \frac{\sin b}{1} \]

   if -9e3 < b

   1. Initial program 87.7%

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

   3. Taylor expanded in b around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-cos.f64 73.3

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

   5. Applied rewrites 73.3%

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

Alternative 14: 51.1% accurate, 1.9× 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 75.9%

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

3. Taylor expanded in b around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cos.f64 55.6

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

5. Applied rewrites 55.6%

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

Alternative 15: 35.1% accurate, 36.7× 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 75.9%

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

3. Taylor expanded in b around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-cos.f64 55.6

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

5. Applied rewrites 55.6%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 35.6%

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

9. Final simplification 35.6%

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

Reproduce

herbie shell --seed 2024238 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))