rsin A (should all be same)

Percentage Accurate: 76.2% → 99.5%

Time: 12.6s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. lift-sin.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 78.3

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 78.3

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

4. Applied egg-rr 78.3%

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

   1. cos-sum N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. unsub-neg N/A

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

   8. lift-neg.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-neg.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. distribute-rgt-neg-in N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-*.f64 99.5

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

6. Applied egg-rr 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

   1. lift-sin.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 78.3

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 78.3

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

4. Applied egg-rr 78.3%

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

   1. cos-sum N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-*.f64 99.5

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 3: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \frac{\sin b}{\cos b}\\ \mathbf{if}\;b \leq -0.0032:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.0017:\\ \;\;\;\;\frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b, b \cdot \left(b \cdot -0.16666666666666666\right), b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (/ (sin b) (cos b)))))
   (if (<= b -0.0032)
     t_0
     (if (<= b 0.0017)
       (* (/ r (cos (+ b a))) (fma b (* b (* b -0.16666666666666666)) b))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * (sin(b) / cos(b));
	double tmp;
	if (b <= -0.0032) {
		tmp = t_0;
	} else if (b <= 0.0017) {
		tmp = (r / cos((b + a))) * fma(b, (b * (b * -0.16666666666666666)), b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * Float64(sin(b) / cos(b)))
	tmp = 0.0
	if (b <= -0.0032)
		tmp = t_0;
	elseif (b <= 0.0017)
		tmp = Float64(Float64(r / cos(Float64(b + a))) * fma(b, Float64(b * Float64(b * -0.16666666666666666)), b));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.0032], t$95$0, If[LessEqual[b, 0.0017], N[(N[(r / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b * N[(b * N[(b * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.00320000000000000015 or 0.00169999999999999991 < b

   1. Initial program 59.7%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 59.8

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 59.8

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

   4. Applied egg-rr 59.8%

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

   5. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-cos.f64 60.2

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

   7. Simplified 60.2%

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

   if -0.00320000000000000015 < b < 0.00169999999999999991

   1. Initial program 98.0%

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

      1. cos-sum N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 99.8

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

   4. Applied egg-rr 99.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. unsub-neg N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. cos-sum N/A

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

      14. lift-+.f64 N/A

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

      15. lift-cos.f64 N/A

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

      16. associate-/l* N/A

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

      17. clear-num N/A

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

   6. Applied egg-rr 98.0%

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

   7. Taylor expanded in b around 0

      \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\left(b \cdot \left(1 + \frac{-1}{6} \cdot {b}^{2}\right)\right)} \]
   8. 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. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 98.1

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

   9. Simplified 98.1%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.0032:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 0.0017:\\ \;\;\;\;\frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b, b \cdot \left(b \cdot -0.16666666666666666\right), b\right)\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.2% 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 78.3%

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

   1. lift-sin.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 78.4

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 78.4

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

4. Applied egg-rr 78.4%

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

5. Final simplification 78.4%

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

Alternative 5: 55.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot r\\ \mathbf{if}\;b \leq -3.1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 4:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) r)))
   (if (<= b -3.1)
     t_0
     (if (<= b 4.0)
       (*
        r
        (/
         (fma
          (fma
           (* b b)
           (fma b (* b -0.0001984126984126984) 0.008333333333333333)
           -0.16666666666666666)
          (* b (* b b))
          b)
         (cos (+ b a))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * r;
	double tmp;
	if (b <= -3.1) {
		tmp = t_0;
	} else if (b <= 4.0) {
		tmp = r * (fma(fma((b * b), fma(b, (b * -0.0001984126984126984), 0.008333333333333333), -0.16666666666666666), (b * (b * b)), b) / cos((b + a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * r)
	tmp = 0.0
	if (b <= -3.1)
		tmp = t_0;
	elseif (b <= 4.0)
		tmp = Float64(r * Float64(fma(fma(Float64(b * b), fma(b, Float64(b * -0.0001984126984126984), 0.008333333333333333), -0.16666666666666666), Float64(b * Float64(b * b)), b) / cos(Float64(b + a))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -3.1], t$95$0, If[LessEqual[b, 4.0], N[(r * N[(N[(N[(N[(b * b), $MachinePrecision] * N[(b * N[(b * -0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.10000000000000009 or 4 < b

   1. Initial program 59.1%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 6.6

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

   5. Simplified 6.6%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 11.6

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

   8. Simplified 11.6%

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

   if -3.10000000000000009 < b < 4

   1. Initial program 98.0%

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

      1. lift-sin.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 98.1

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 98.1

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

   4. Applied egg-rr 98.1%

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

   5. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{b \cdot \left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)}}{\cos \left(b + a\right)} \cdot r \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{b \cdot \color{blue}{\left({b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) + 1\right)}}{\cos \left(b + a\right)} \cdot r \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left({b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right) \cdot b + 1 \cdot b}}{\cos \left(b + a\right)} \cdot r \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) \cdot {b}^{2}\right)} \cdot b + 1 \cdot b}{\cos \left(b + a\right)} \cdot r \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) \cdot \left({b}^{2} \cdot b\right)} + 1 \cdot b}{\cos \left(b + a\right)} \cdot r \]

      5. unpow2 N/A

         \[\leadsto \frac{\left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot b\right) + 1 \cdot b}{\cos \left(b + a\right)} \cdot r \]

      6. unpow3 N/A

         \[\leadsto \frac{\left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{{b}^{3}} + 1 \cdot b}{\cos \left(b + a\right)} \cdot r \]

      7. *-lft-identity N/A

         \[\leadsto \frac{\left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) \cdot {b}^{3} + \color{blue}{b}}{\cos \left(b + a\right)} \cdot r \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}, {b}^{3}, b\right)}}{\cos \left(b + a\right)} \cdot r \]

   7. Simplified 97.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}}{\cos \left(b + a\right)} \cdot r \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.1:\\ \;\;\;\;\sin b \cdot r\\ \mathbf{elif}\;b \leq 4:\\ \;\;\;\;r \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b \cdot -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), b \cdot \left(b \cdot b\right), b\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot r\\ \mathbf{if}\;b \leq -3.95:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 18000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, r \cdot \mathsf{fma}\left(b, b \cdot 0.008333333333333333, -0.16666666666666666\right), r\right) \cdot \frac{b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) r)))
   (if (<= b -3.95)
     t_0
     (if (<= b 18000000.0)
       (*
        (fma
         (* b b)
         (* r (fma b (* b 0.008333333333333333) -0.16666666666666666))
         r)
        (/ b (cos (+ b a))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * r;
	double tmp;
	if (b <= -3.95) {
		tmp = t_0;
	} else if (b <= 18000000.0) {
		tmp = fma((b * b), (r * fma(b, (b * 0.008333333333333333), -0.16666666666666666)), r) * (b / cos((b + a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * r)
	tmp = 0.0
	if (b <= -3.95)
		tmp = t_0;
	elseif (b <= 18000000.0)
		tmp = Float64(fma(Float64(b * b), Float64(r * fma(b, Float64(b * 0.008333333333333333), -0.16666666666666666)), r) * Float64(b / cos(Float64(b + a))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -3.95], t$95$0, If[LessEqual[b, 18000000.0], N[(N[(N[(b * b), $MachinePrecision] * N[(r * N[(b * N[(b * 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + r), $MachinePrecision] * N[(b / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.9500000000000002 or 1.8e7 < b

   1. Initial program 60.0%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 6.6

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

   5. Simplified 6.6%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 11.8

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

   8. Simplified 11.8%

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

   if -3.9500000000000002 < b < 1.8e7

   1. Initial program 96.5%

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

   3. Taylor expanded in b around 0

      \[\leadsto \frac{\color{blue}{b \cdot \left(r + {b}^{2} \cdot \left(\frac{-1}{6} \cdot r + \frac{1}{120} \cdot \left({b}^{2} \cdot r\right)\right)\right)}}{\cos \left(a + b\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{b \cdot \left(r + {b}^{2} \cdot \left(\frac{-1}{6} \cdot r + \frac{1}{120} \cdot \left({b}^{2} \cdot r\right)\right)\right)}}{\cos \left(a + b\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{b \cdot \color{blue}{\left({b}^{2} \cdot \left(\frac{-1}{6} \cdot r + \frac{1}{120} \cdot \left({b}^{2} \cdot r\right)\right) + r\right)}}{\cos \left(a + b\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{b \cdot \color{blue}{\mathsf{fma}\left({b}^{2}, \frac{-1}{6} \cdot r + \frac{1}{120} \cdot \left({b}^{2} \cdot r\right), r\right)}}{\cos \left(a + b\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{b \cdot \mathsf{fma}\left(\color{blue}{b \cdot b}, \frac{-1}{6} \cdot r + \frac{1}{120} \cdot \left({b}^{2} \cdot r\right), r\right)}{\cos \left(a + b\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{b \cdot \mathsf{fma}\left(\color{blue}{b \cdot b}, \frac{-1}{6} \cdot r + \frac{1}{120} \cdot \left({b}^{2} \cdot r\right), r\right)}{\cos \left(a + b\right)} \]

      6. associate-*r* N/A

         \[\leadsto \frac{b \cdot \mathsf{fma}\left(b \cdot b, \frac{-1}{6} \cdot r + \color{blue}{\left(\frac{1}{120} \cdot {b}^{2}\right) \cdot r}, r\right)}{\cos \left(a + b\right)} \]

      7. distribute-rgt-out N/A

         \[\leadsto \frac{b \cdot \mathsf{fma}\left(b \cdot b, \color{blue}{r \cdot \left(\frac{-1}{6} + \frac{1}{120} \cdot {b}^{2}\right)}, r\right)}{\cos \left(a + b\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{b \cdot \mathsf{fma}\left(b \cdot b, r \cdot \color{blue}{\left(\frac{1}{120} \cdot {b}^{2} + \frac{-1}{6}\right)}, r\right)}{\cos \left(a + b\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{b \cdot \mathsf{fma}\left(b \cdot b, r \cdot \left(\frac{1}{120} \cdot {b}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right), r\right)}{\cos \left(a + b\right)} \]

      10. sub-neg N/A

          \[\leadsto \frac{b \cdot \mathsf{fma}\left(b \cdot b, r \cdot \color{blue}{\left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)}, r\right)}{\cos \left(a + b\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{b \cdot \mathsf{fma}\left(b \cdot b, \color{blue}{r \cdot \left(\frac{1}{120} \cdot {b}^{2} - \frac{1}{6}\right)}, r\right)}{\cos \left(a + b\right)} \]

      12. sub-neg N/A

          \[\leadsto \frac{b \cdot \mathsf{fma}\left(b \cdot b, r \cdot \color{blue}{\left(\frac{1}{120} \cdot {b}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, r\right)}{\cos \left(a + b\right)} \]

      13. *-commutative N/A

          \[\leadsto \frac{b \cdot \mathsf{fma}\left(b \cdot b, r \cdot \left(\color{blue}{{b}^{2} \cdot \frac{1}{120}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right), r\right)}{\cos \left(a + b\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{b \cdot \mathsf{fma}\left(b \cdot b, r \cdot \left({b}^{2} \cdot \frac{1}{120} + \color{blue}{\frac{-1}{6}}\right), r\right)}{\cos \left(a + b\right)} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{b \cdot \mathsf{fma}\left(b \cdot b, r \cdot \color{blue}{\mathsf{fma}\left({b}^{2}, \frac{1}{120}, \frac{-1}{6}\right)}, r\right)}{\cos \left(a + b\right)} \]

      16. unpow2 N/A

          \[\leadsto \frac{b \cdot \mathsf{fma}\left(b \cdot b, r \cdot \mathsf{fma}\left(\color{blue}{b \cdot b}, \frac{1}{120}, \frac{-1}{6}\right), r\right)}{\cos \left(a + b\right)} \]

      17. lower-*.f64 96.0

          \[\leadsto \frac{b \cdot \mathsf{fma}\left(b \cdot b, r \cdot \mathsf{fma}\left(\color{blue}{b \cdot b}, 0.008333333333333333, -0.16666666666666666\right), r\right)}{\cos \left(a + b\right)} \]

   5. Simplified 96.0%

      \[\leadsto \frac{\color{blue}{b \cdot \mathsf{fma}\left(b \cdot b, r \cdot \mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), r\right)}}{\cos \left(a + b\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{b \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(r \cdot \left(\left(b \cdot b\right) \cdot \frac{1}{120} + \frac{-1}{6}\right)\right) + r\right)}{\cos \left(a + b\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{b \cdot \left(\left(b \cdot b\right) \cdot \left(r \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \frac{1}{120} + \frac{-1}{6}\right)\right) + r\right)}{\cos \left(a + b\right)} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{b \cdot \left(\left(b \cdot b\right) \cdot \left(r \cdot \color{blue}{\mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right)}\right) + r\right)}{\cos \left(a + b\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{b \cdot \left(\left(b \cdot b\right) \cdot \color{blue}{\left(r \cdot \mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right)\right)} + r\right)}{\cos \left(a + b\right)} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{b \cdot \color{blue}{\mathsf{fma}\left(b \cdot b, r \cdot \mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), r\right)}}{\cos \left(a + b\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b \cdot b, r \cdot \mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), r\right) \cdot b}}{\cos \left(a + b\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(b \cdot b, r \cdot \mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), r\right) \cdot b}{\cos \color{blue}{\left(b + a\right)}} \]

      8. lift-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b \cdot b, r \cdot \mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), r\right) \cdot b}{\cos \color{blue}{\left(b + a\right)}} \]

      9. lift-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b \cdot b, r \cdot \mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), r\right) \cdot b}{\color{blue}{\cos \left(b + a\right)}} \]

      10. associate-/l* N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, r \cdot \mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), r\right) \cdot \frac{b}{\cos \left(b + a\right)}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, r \cdot \mathsf{fma}\left(b \cdot b, \frac{1}{120}, \frac{-1}{6}\right), r\right) \cdot \frac{b}{\cos \left(b + a\right)}} \]

   7. Applied egg-rr 96.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot b, r \cdot \mathsf{fma}\left(b, b \cdot 0.008333333333333333, -0.16666666666666666\right), r\right) \cdot \frac{b}{\cos \left(b + a\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.95:\\ \;\;\;\;\sin b \cdot r\\ \mathbf{elif}\;b \leq 18000000:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, r \cdot \mathsf{fma}\left(b, b \cdot 0.008333333333333333, -0.16666666666666666\right), r\right) \cdot \frac{b}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot r\\ \mathbf{if}\;b \leq -3.1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 4:\\ \;\;\;\;\frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b, b \cdot \left(b \cdot -0.16666666666666666\right), b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) r)))
   (if (<= b -3.1)
     t_0
     (if (<= b 4.0)
       (* (/ r (cos (+ b a))) (fma b (* b (* b -0.16666666666666666)) b))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * r;
	double tmp;
	if (b <= -3.1) {
		tmp = t_0;
	} else if (b <= 4.0) {
		tmp = (r / cos((b + a))) * fma(b, (b * (b * -0.16666666666666666)), b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * r)
	tmp = 0.0
	if (b <= -3.1)
		tmp = t_0;
	elseif (b <= 4.0)
		tmp = Float64(Float64(r / cos(Float64(b + a))) * fma(b, Float64(b * Float64(b * -0.16666666666666666)), b));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -3.1], t$95$0, If[LessEqual[b, 4.0], N[(N[(r / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b * N[(b * N[(b * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.10000000000000009 or 4 < b

   1. Initial program 59.1%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 6.6

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

   5. Simplified 6.6%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 11.6

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

   8. Simplified 11.6%

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

   if -3.10000000000000009 < b < 4

   1. Initial program 98.0%

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

      1. cos-sum N/A

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 99.8

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

   4. Applied egg-rr 99.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. unsub-neg N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. cos-sum N/A

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

      14. lift-+.f64 N/A

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

      15. lift-cos.f64 N/A

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

      16. associate-/l* N/A

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

      17. clear-num N/A

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

   6. Applied egg-rr 98.0%

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

   7. Taylor expanded in b around 0

      \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\left(b \cdot \left(1 + \frac{-1}{6} \cdot {b}^{2}\right)\right)} \]
   8. 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. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 97.2

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

   9. Simplified 97.2%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.1:\\ \;\;\;\;\sin b \cdot r\\ \mathbf{elif}\;b \leq 4:\\ \;\;\;\;\frac{r}{\cos \left(b + a\right)} \cdot \mathsf{fma}\left(b, b \cdot \left(b \cdot -0.16666666666666666\right), b\right)\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot r\\ \mathbf{if}\;b \leq -7.4 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 18500000:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) r)))
   (if (<= b -7.4e+19)
     t_0
     (if (<= b 18500000.0) (/ (* b r) (cos (+ b a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * r;
	double tmp;
	if (b <= -7.4e+19) {
		tmp = t_0;
	} else if (b <= 18500000.0) {
		tmp = (b * r) / cos((b + a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(b) * r
    if (b <= (-7.4d+19)) then
        tmp = t_0
    else if (b <= 18500000.0d0) then
        tmp = (b * r) / cos((b + a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.sin(b) * r;
	double tmp;
	if (b <= -7.4e+19) {
		tmp = t_0;
	} else if (b <= 18500000.0) {
		tmp = (b * r) / Math.cos((b + a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) * r
	tmp = 0
	if b <= -7.4e+19:
		tmp = t_0
	elif b <= 18500000.0:
		tmp = (b * r) / math.cos((b + a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * r)
	tmp = 0.0
	if (b <= -7.4e+19)
		tmp = t_0;
	elseif (b <= 18500000.0)
		tmp = Float64(Float64(b * r) / cos(Float64(b + a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) * r;
	tmp = 0.0;
	if (b <= -7.4e+19)
		tmp = t_0;
	elseif (b <= 18500000.0)
		tmp = (b * r) / cos((b + a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -7.4e+19], t$95$0, If[LessEqual[b, 18500000.0], N[(N[(b * r), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -7.4e19 or 1.85e7 < b

   1. Initial program 58.8%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 6.6

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

   5. Simplified 6.6%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 11.9

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

   8. Simplified 11.9%

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

   if -7.4e19 < b < 1.85e7

   1. Initial program 96.6%

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

   3. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 92.8

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

   5. Simplified 92.8%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.4 \cdot 10^{+19}:\\ \;\;\;\;\sin b \cdot r\\ \mathbf{elif}\;b \leq 18500000:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot r\\ \mathbf{if}\;b \leq -4.7:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 18500000:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) r)))
   (if (<= b -4.7) t_0 (if (<= b 18500000.0) (* b (/ r (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * r;
	double tmp;
	if (b <= -4.7) {
		tmp = t_0;
	} else if (b <= 18500000.0) {
		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 = sin(b) * r
    if (b <= (-4.7d0)) then
        tmp = t_0
    else if (b <= 18500000.0d0) 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 = Math.sin(b) * r;
	double tmp;
	if (b <= -4.7) {
		tmp = t_0;
	} else if (b <= 18500000.0) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) * r
	tmp = 0
	if b <= -4.7:
		tmp = t_0
	elif b <= 18500000.0:
		tmp = b * (r / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * r)
	tmp = 0.0
	if (b <= -4.7)
		tmp = t_0;
	elseif (b <= 18500000.0)
		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 = sin(b) * r;
	tmp = 0.0;
	if (b <= -4.7)
		tmp = t_0;
	elseif (b <= 18500000.0)
		tmp = b * (r / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[b, -4.7], t$95$0, If[LessEqual[b, 18500000.0], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.70000000000000018 or 1.85e7 < b

   1. Initial program 60.0%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 6.6

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

   5. Simplified 6.6%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 11.8

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

   8. Simplified 11.8%

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

   if -4.70000000000000018 < b < 1.85e7

   1. Initial program 96.5%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   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. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 95.3

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

   5. Simplified 95.3%

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

      1. lift-cos.f64 N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 95.4

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

   7. Applied egg-rr 95.4%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.7:\\ \;\;\;\;\sin b \cdot r\\ \mathbf{elif}\;b \leq 18500000:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

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

3. Taylor expanded in b around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 51.5

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

5. Simplified 51.5%

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

6. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 38.3

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

8. Simplified 38.3%

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

9. Final simplification 38.3%

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

Alternative 11: 34.7% accurate, 36.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.3%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
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. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 49.4

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

5. Simplified 49.4%

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

6. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 34.1

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

8. Simplified 34.1%

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

9. Final simplification 34.1%

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

Reproduce

herbie shell --seed 2024208 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))