rsin A (should all be same)

Percentage Accurate: 76.7% → 99.5%

Time: 15.3s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.7% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. cos-lowering-cos.f64 N/A

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

   7. +-commutative N/A

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

   8. +-lowering-+.f64 N/A

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

   9. sin-lowering-sin.f64 74.1

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

4. Applied egg-rr 74.1%

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

   1. cos-sum N/A

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

   2. *-commutative N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. sub0-neg N/A

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

   5. +-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. --lowering--.f64 N/A

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

   8. sin-lowering-sin.f64 N/A

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

   9. sin-lowering-sin.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. cos-lowering-cos.f64 N/A

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

   12. cos-lowering-cos.f64 99.6

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

6. Applied egg-rr 99.6%

   \[\leadsto \frac{r}{\color{blue}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)}} \cdot \sin b \]
7. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 N/A

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

   3. sin-lowering-sin.f64 99.6

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

8. Applied egg-rr 99.6%

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

9. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. cos-lowering-cos.f64 N/A

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

   7. +-commutative N/A

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

   8. +-lowering-+.f64 N/A

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

   9. sin-lowering-sin.f64 74.1

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

4. Applied egg-rr 74.1%

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

   1. cos-sum N/A

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

   2. *-commutative N/A

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

   3. --lowering--.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. cos-lowering-cos.f64 N/A

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

   6. cos-lowering-cos.f64 N/A

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

   7. *-lowering-*.f64 N/A

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

   8. sin-lowering-sin.f64 N/A

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

   9. sin-lowering-sin.f64 99.5

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 3: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.076:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{elif}\;b \leq 0.07:\\ \;\;\;\;r \cdot \frac{b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -0.076)
   (* (sin b) (/ r (cos b)))
   (if (<= b 0.07)
     (*
      r
      (/
       (*
        b
        (fma
         (* b b)
         (fma
          b
          (* b (fma (* b b) -0.0001984126984126984 0.008333333333333333))
          -0.16666666666666666)
         1.0))
       (cos (+ b a))))
     (/ (* r (sin b)) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -0.076) {
		tmp = sin(b) * (r / cos(b));
	} else if (b <= 0.07) {
		tmp = r * ((b * fma((b * b), fma(b, (b * fma((b * b), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), 1.0)) / cos((b + a)));
	} else {
		tmp = (r * sin(b)) / cos(b);
	}
	return tmp;
}

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -0.076)
		tmp = Float64(sin(b) * Float64(r / cos(b)));
	elseif (b <= 0.07)
		tmp = Float64(r * Float64(Float64(b * fma(Float64(b * b), fma(b, Float64(b * fma(Float64(b * b), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), 1.0)) / cos(Float64(b + a))));
	else
		tmp = Float64(Float64(r * sin(b)) / cos(b));
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -0.076], N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 0.07], N[(r * N[(N[(b * N[(N[(b * b), $MachinePrecision] * N[(b * N[(b * N[(N[(b * b), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -0.0759999999999999981

   1. Initial program 56.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 N/A

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

      9. sin-lowering-sin.f64 56.7

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

   4. Applied egg-rr 56.7%

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

   5. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. cos-lowering-cos.f64 58.2

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

   7. Simplified 58.2%

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

   if -0.0759999999999999981 < b < 0.070000000000000007

   1. Initial program 97.5%

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

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-in N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. neg-sub0 N/A

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

      8. --lowering--.f64 N/A

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

      9. sin-lowering-sin.f64 N/A

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

      10. sin-lowering-sin.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. cos-lowering-cos.f64 N/A

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

      13. cos-lowering-cos.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)\right)}}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)\right)}}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \color{blue}{\left({b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \color{blue}{\mathsf{fma}\left({b}^{2}, {b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}, 1\right)}\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(\color{blue}{b \cdot b}, {b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}, 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(\color{blue}{b \cdot b}, {b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}, 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      6. sub-neg N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \color{blue}{{b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, {b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) + \color{blue}{\frac{-1}{6}}, 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \color{blue}{\mathsf{fma}\left({b}^{2}, \frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}, \frac{-1}{6}\right)}, 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(\color{blue}{b \cdot b}, \frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}, \frac{-1}{6}\right), 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(\color{blue}{b \cdot b}, \frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}, \frac{-1}{6}\right), 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, \color{blue}{\frac{-1}{5040} \cdot {b}^{2} + \frac{1}{120}}, \frac{-1}{6}\right), 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, \color{blue}{{b}^{2} \cdot \frac{-1}{5040}} + \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, \color{blue}{\mathsf{fma}\left({b}^{2}, \frac{-1}{5040}, \frac{1}{120}\right)}, \frac{-1}{6}\right), 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(\color{blue}{b \cdot b}, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      15. *-lowering-*.f64 99.8

          \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(\color{blue}{b \cdot b}, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

   7. Simplified 99.8%

      \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)}}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{r \cdot \frac{b \cdot \left(\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \frac{-1}{5040} + \frac{1}{120}\right) + \frac{-1}{6}\right) + 1\right)}{\left(0 - \sin b\right) \cdot \sin a + \cos a \cdot \cos b}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{b \cdot \left(\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \frac{-1}{5040} + \frac{1}{120}\right) + \frac{-1}{6}\right) + 1\right)}{\left(0 - \sin b\right) \cdot \sin a + \cos a \cdot \cos b} \cdot r} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{b \cdot \left(\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \frac{-1}{5040} + \frac{1}{120}\right) + \frac{-1}{6}\right) + 1\right)}{\left(0 - \sin b\right) \cdot \sin a + \cos a \cdot \cos b} \cdot r} \]

   9. Applied egg-rr 97.6%

      \[\leadsto \color{blue}{\frac{b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{\cos \left(b + a\right)} \cdot r} \]

   if 0.070000000000000007 < b

   1. Initial program 41.9%

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

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

      2. *-lowering-*.f64 N/A

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

      3. sin-lowering-sin.f64 N/A

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

      4. cos-lowering-cos.f64 41.7

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

   5. Simplified 41.7%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.076:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{elif}\;b \leq 0.07:\\ \;\;\;\;r \cdot \frac{b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot \frac{r}{\cos b}\\ \mathbf{if}\;b \leq -0.235:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.095:\\ \;\;\;\;r \cdot \frac{b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (/ r (cos b)))))
   (if (<= b -0.235)
     t_0
     (if (<= b 0.095)
       (*
        r
        (/
         (*
          b
          (fma
           (* b b)
           (fma
            b
            (* b (fma (* b b) -0.0001984126984126984 0.008333333333333333))
            -0.16666666666666666)
           1.0))
         (cos (+ b a))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * (r / cos(b));
	double tmp;
	if (b <= -0.235) {
		tmp = t_0;
	} else if (b <= 0.095) {
		tmp = r * ((b * fma((b * b), fma(b, (b * fma((b * b), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), 1.0)) / cos((b + a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * Float64(r / cos(b)))
	tmp = 0.0
	if (b <= -0.235)
		tmp = t_0;
	elseif (b <= 0.095)
		tmp = Float64(r * Float64(Float64(b * fma(Float64(b * b), fma(b, Float64(b * fma(Float64(b * b), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), 1.0)) / 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] * N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -0.235], t$95$0, If[LessEqual[b, 0.095], N[(r * N[(N[(b * N[(N[(b * b), $MachinePrecision] * N[(b * N[(b * N[(N[(b * b), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -0.23499999999999999 or 0.095000000000000001 < b

   1. Initial program 49.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 N/A

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

      9. sin-lowering-sin.f64 49.9

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

   4. Applied egg-rr 49.9%

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

   5. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. cos-lowering-cos.f64 50.6

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

   7. Simplified 50.6%

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

   if -0.23499999999999999 < b < 0.095000000000000001

   1. Initial program 97.5%

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

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-in N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. neg-sub0 N/A

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

      8. --lowering--.f64 N/A

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

      9. sin-lowering-sin.f64 N/A

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

      10. sin-lowering-sin.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. cos-lowering-cos.f64 N/A

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

      13. cos-lowering-cos.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)\right)}}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)\right)}}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \color{blue}{\left({b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \color{blue}{\mathsf{fma}\left({b}^{2}, {b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}, 1\right)}\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(\color{blue}{b \cdot b}, {b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}, 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(\color{blue}{b \cdot b}, {b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}, 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      6. sub-neg N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \color{blue}{{b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, {b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) + \color{blue}{\frac{-1}{6}}, 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \color{blue}{\mathsf{fma}\left({b}^{2}, \frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}, \frac{-1}{6}\right)}, 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(\color{blue}{b \cdot b}, \frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}, \frac{-1}{6}\right), 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(\color{blue}{b \cdot b}, \frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}, \frac{-1}{6}\right), 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, \color{blue}{\frac{-1}{5040} \cdot {b}^{2} + \frac{1}{120}}, \frac{-1}{6}\right), 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, \color{blue}{{b}^{2} \cdot \frac{-1}{5040}} + \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, \color{blue}{\mathsf{fma}\left({b}^{2}, \frac{-1}{5040}, \frac{1}{120}\right)}, \frac{-1}{6}\right), 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(\color{blue}{b \cdot b}, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      15. *-lowering-*.f64 99.8

          \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(\color{blue}{b \cdot b}, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

   7. Simplified 99.8%

      \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)}}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{r \cdot \frac{b \cdot \left(\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \frac{-1}{5040} + \frac{1}{120}\right) + \frac{-1}{6}\right) + 1\right)}{\left(0 - \sin b\right) \cdot \sin a + \cos a \cdot \cos b}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{b \cdot \left(\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \frac{-1}{5040} + \frac{1}{120}\right) + \frac{-1}{6}\right) + 1\right)}{\left(0 - \sin b\right) \cdot \sin a + \cos a \cdot \cos b} \cdot r} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{b \cdot \left(\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \frac{-1}{5040} + \frac{1}{120}\right) + \frac{-1}{6}\right) + 1\right)}{\left(0 - \sin b\right) \cdot \sin a + \cos a \cdot \cos b} \cdot r} \]

   9. Applied egg-rr 97.6%

      \[\leadsto \color{blue}{\frac{b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{\cos \left(b + a\right)} \cdot r} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.235:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{elif}\;b \leq 0.095:\\ \;\;\;\;r \cdot \frac{b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.7% 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 74.1%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. cos-lowering-cos.f64 N/A

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

   7. +-commutative N/A

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

   8. +-lowering-+.f64 N/A

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

   9. sin-lowering-sin.f64 74.1

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

4. Applied egg-rr 74.1%

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

5. Final simplification 74.1%

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

Alternative 6: 55.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. cos-lowering-cos.f64 N/A

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

   7. +-commutative N/A

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

   8. +-lowering-+.f64 N/A

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

   9. sin-lowering-sin.f64 74.1

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

4. Applied egg-rr 74.1%

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

5. Taylor expanded in b around 0

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

   1. /-lowering-/.f64 N/A

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

   2. cos-lowering-cos.f64 54.6

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

7. Simplified 54.6%

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

8. Final simplification 54.6%

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

Alternative 7: 55.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -5.2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.69:\\ \;\;\;\;r \cdot \frac{b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -5.2)
     t_0
     (if (<= b 0.69)
       (*
        r
        (/
         (*
          b
          (fma
           (* b b)
           (fma
            b
            (* b (fma (* b b) -0.0001984126984126984 0.008333333333333333))
            -0.16666666666666666)
           1.0))
         (cos (+ b a))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -5.2) {
		tmp = t_0;
	} else if (b <= 0.69) {
		tmp = r * ((b * fma((b * b), fma(b, (b * fma((b * b), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), 1.0)) / cos((b + a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -5.2)
		tmp = t_0;
	elseif (b <= 0.69)
		tmp = Float64(r * Float64(Float64(b * fma(Float64(b * b), fma(b, Float64(b * fma(Float64(b * b), -0.0001984126984126984, 0.008333333333333333)), -0.16666666666666666), 1.0)) / cos(Float64(b + a))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -5.2], t$95$0, If[LessEqual[b, 0.69], N[(r * N[(N[(b * N[(N[(b * b), $MachinePrecision] * N[(b * N[(b * N[(N[(b * b), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -5.20000000000000018 or 0.68999999999999995 < b

   1. Initial program 49.5%

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

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

      4. cos-lowering-cos.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sin-lowering-sin.f64 6.5

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

   5. Simplified 6.5%

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

   6. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 11.4

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

   8. Simplified 11.4%

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

   if -5.20000000000000018 < b < 0.68999999999999995

   1. Initial program 97.5%

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

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-in N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. neg-sub0 N/A

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

      8. --lowering--.f64 N/A

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

      9. sin-lowering-sin.f64 N/A

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

      10. sin-lowering-sin.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. cos-lowering-cos.f64 N/A

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

      13. cos-lowering-cos.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in b around 0

      \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)\right)}}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \left(1 + {b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right)\right)\right)}}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \color{blue}{\left({b}^{2} \cdot \left({b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}\right) + 1\right)}\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \color{blue}{\mathsf{fma}\left({b}^{2}, {b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}, 1\right)}\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(\color{blue}{b \cdot b}, {b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}, 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(\color{blue}{b \cdot b}, {b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) - \frac{1}{6}, 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      6. sub-neg N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \color{blue}{{b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}, 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, {b}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}\right) + \color{blue}{\frac{-1}{6}}, 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \color{blue}{\mathsf{fma}\left({b}^{2}, \frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}, \frac{-1}{6}\right)}, 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(\color{blue}{b \cdot b}, \frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}, \frac{-1}{6}\right), 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(\color{blue}{b \cdot b}, \frac{1}{120} + \frac{-1}{5040} \cdot {b}^{2}, \frac{-1}{6}\right), 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, \color{blue}{\frac{-1}{5040} \cdot {b}^{2} + \frac{1}{120}}, \frac{-1}{6}\right), 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      12. *-commutative N/A

          \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, \color{blue}{{b}^{2} \cdot \frac{-1}{5040}} + \frac{1}{120}, \frac{-1}{6}\right), 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, \color{blue}{\mathsf{fma}\left({b}^{2}, \frac{-1}{5040}, \frac{1}{120}\right)}, \frac{-1}{6}\right), 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(\color{blue}{b \cdot b}, \frac{-1}{5040}, \frac{1}{120}\right), \frac{-1}{6}\right), 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

      15. *-lowering-*.f64 99.5

          \[\leadsto \frac{r \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(\color{blue}{b \cdot b}, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]

   7. Simplified 99.5%

      \[\leadsto \frac{r \cdot \color{blue}{\left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)}}{\mathsf{fma}\left(0 - \sin b, \sin a, \cos a \cdot \cos b\right)} \]
   8. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{r \cdot \frac{b \cdot \left(\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \frac{-1}{5040} + \frac{1}{120}\right) + \frac{-1}{6}\right) + 1\right)}{\left(0 - \sin b\right) \cdot \sin a + \cos a \cdot \cos b}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{b \cdot \left(\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \frac{-1}{5040} + \frac{1}{120}\right) + \frac{-1}{6}\right) + 1\right)}{\left(0 - \sin b\right) \cdot \sin a + \cos a \cdot \cos b} \cdot r} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{b \cdot \left(\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \frac{-1}{5040} + \frac{1}{120}\right) + \frac{-1}{6}\right) + 1\right)}{\left(0 - \sin b\right) \cdot \sin a + \cos a \cdot \cos b} \cdot r} \]

   9. Applied egg-rr 97.4%

      \[\leadsto \color{blue}{\frac{b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{\cos \left(b + a\right)} \cdot r} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{elif}\;b \leq 0.69:\\ \;\;\;\;r \cdot \frac{b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \sin b\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -4.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.69:\\ \;\;\;\;\frac{r}{\cos \left(b + a\right)} \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -4.6)
     t_0
     (if (<= b 0.69)
       (*
        (/ r (cos (+ b a)))
        (*
         b
         (fma
          (* b b)
          (fma
           (* b b)
           (fma (* b b) -0.0001984126984126984 0.008333333333333333)
           -0.16666666666666666)
          1.0)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -4.6) {
		tmp = t_0;
	} else if (b <= 0.69) {
		tmp = (r / cos((b + a))) * (b * fma((b * b), fma((b * b), fma((b * b), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -4.6)
		tmp = t_0;
	elseif (b <= 0.69)
		tmp = Float64(Float64(r / cos(Float64(b + a))) * Float64(b * fma(Float64(b * b), fma(Float64(b * b), fma(Float64(b * b), -0.0001984126984126984, 0.008333333333333333), -0.16666666666666666), 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.6], t$95$0, If[LessEqual[b, 0.69], N[(N[(r / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b * N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.5999999999999996 or 0.68999999999999995 < b

   1. Initial program 49.5%

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

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

      4. cos-lowering-cos.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sin-lowering-sin.f64 6.5

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

   5. Simplified 6.5%

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

   6. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 11.4

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

   8. Simplified 11.4%

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

   if -4.5999999999999996 < b < 0.68999999999999995

   1. Initial program 97.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 N/A

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

      9. sin-lowering-sin.f64 97.6

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

   4. Applied egg-rr 97.6%

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

   5. Taylor expanded in b around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 97.4

          \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(\color{blue}{b \cdot b}, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right) \]

   7. Simplified 97.4%

      \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b \cdot b, -0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right), 1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 55.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -4.2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.69:\\ \;\;\;\;r \cdot \frac{b \cdot \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), 1\right)}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -4.2)
     t_0
     (if (<= b 0.69)
       (*
        r
        (/
         (*
          b
          (fma
           b
           (* b (fma (* b b) 0.008333333333333333 -0.16666666666666666))
           1.0))
         (cos (+ b a))))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -4.2) {
		tmp = t_0;
	} else if (b <= 0.69) {
		tmp = r * ((b * fma(b, (b * fma((b * b), 0.008333333333333333, -0.16666666666666666)), 1.0)) / cos((b + a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -4.2)
		tmp = t_0;
	elseif (b <= 0.69)
		tmp = Float64(r * Float64(Float64(b * fma(b, Float64(b * fma(Float64(b * b), 0.008333333333333333, -0.16666666666666666)), 1.0)) / cos(Float64(b + a))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.2], t$95$0, If[LessEqual[b, 0.69], N[(r * N[(N[(b * N[(b * N[(b * N[(N[(b * b), $MachinePrecision] * 0.008333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.20000000000000018 or 0.68999999999999995 < b

   1. Initial program 49.5%

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

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

      4. cos-lowering-cos.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sin-lowering-sin.f64 6.5

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

   5. Simplified 6.5%

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

   6. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 11.4

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

   8. Simplified 11.4%

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

   if -4.20000000000000018 < b < 0.68999999999999995

   1. Initial program 97.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 N/A

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

      9. sin-lowering-sin.f64 97.6

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

   4. Applied egg-rr 97.6%

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

   5. Taylor expanded in b around 0

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. metadata-eval N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 97.2

          \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, \color{blue}{b \cdot 0.008333333333333333}, -0.16666666666666666\right), 1\right)\right) \]

   7. Simplified 97.2%

      \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\left(b \cdot \mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(b, b \cdot 0.008333333333333333, -0.16666666666666666\right), 1\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. cos-lowering-cos.f64 N/A

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

      13. +-lowering-+.f64 97.2

          \[\leadsto r \cdot \frac{b \cdot \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), 1\right)}{\cos \color{blue}{\left(b + a\right)}} \]

   9. Applied egg-rr 97.2%

      \[\leadsto \color{blue}{r \cdot \frac{b \cdot \mathsf{fma}\left(b, b \cdot \mathsf{fma}\left(b \cdot b, 0.008333333333333333, -0.16666666666666666\right), 1\right)}{\cos \left(b + a\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 55.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \sin b\\ \mathbf{if}\;b \leq -3.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 0.69:\\ \;\;\;\;\frac{r}{\cos \left(b + a\right)} \cdot \left(b \cdot \mathsf{fma}\left(b, b \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (sin b))))
   (if (<= b -3.5)
     t_0
     (if (<= b 0.69)
       (* (/ r (cos (+ b a))) (* b (fma b (* b -0.16666666666666666) 1.0)))
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * sin(b);
	double tmp;
	if (b <= -3.5) {
		tmp = t_0;
	} else if (b <= 0.69) {
		tmp = (r / cos((b + a))) * (b * fma(b, (b * -0.16666666666666666), 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(r * sin(b))
	tmp = 0.0
	if (b <= -3.5)
		tmp = t_0;
	elseif (b <= 0.69)
		tmp = Float64(Float64(r / cos(Float64(b + a))) * Float64(b * fma(b, Float64(b * -0.16666666666666666), 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.5], t$95$0, If[LessEqual[b, 0.69], N[(N[(r / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(b * N[(b * N[(b * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.5 or 0.68999999999999995 < b

   1. Initial program 49.5%

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

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

      4. cos-lowering-cos.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sin-lowering-sin.f64 6.5

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

   5. Simplified 6.5%

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

   6. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 11.4

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

   8. Simplified 11.4%

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

   if -3.5 < b < 0.68999999999999995

   1. Initial program 97.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 N/A

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

      9. sin-lowering-sin.f64 97.6

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

   4. Applied egg-rr 97.6%

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

   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. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 96.8

         \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \left(b \cdot \mathsf{fma}\left(b, \color{blue}{b \cdot -0.16666666666666666}, 1\right)\right) \]

   7. Simplified 96.8%

      \[\leadsto \frac{r}{\cos \left(b + a\right)} \cdot \color{blue}{\left(b \cdot \mathsf{fma}\left(b, b \cdot -0.16666666666666666, 1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 53.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -4.5) (* r (sin b)) (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -4.5) {
		tmp = r * sin(b);
	} else {
		tmp = b * (r / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-4.5d0)) then
        tmp = r * sin(b)
    else
        tmp = b * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -4.5) {
		tmp = r * Math.sin(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -4.5:
		tmp = r * math.sin(b)
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -4.5)
		tmp = Float64(r * sin(b));
	else
		tmp = Float64(b * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -4.5)
		tmp = r * sin(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -4.5], N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.5

   1. Initial program 56.0%

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

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

      4. cos-lowering-cos.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sin-lowering-sin.f64 6.5

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

   5. Simplified 6.5%

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

   6. Taylor expanded in a around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 12.2

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

   8. Simplified 12.2%

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

   if -4.5 < b

   1. Initial program 80.5%

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

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

      4. cos-lowering-cos.f64 68.0

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

   5. Simplified 68.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. cos-lowering-cos.f64 68.1

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

   7. Applied egg-rr 68.1%

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

Alternative 12: 39.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

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

   4. cos-lowering-cos.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. sin-lowering-sin.f64 53.0

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

5. Simplified 53.0%

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

6. Taylor expanded in a around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sin-lowering-sin.f64 39.5

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

8. Simplified 39.5%

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

Alternative 13: 35.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ b \cdot \left(r \cdot \frac{1}{\mathsf{fma}\left(b, b \cdot -0.5, 1\right)}\right) \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* b (* r (/ 1.0 (fma b (* b -0.5) 1.0)))))

C

double code(double r, double a, double b) {
	return b * (r * (1.0 / fma(b, (b * -0.5), 1.0)));
}

Julia

function code(r, a, b)
	return Float64(b * Float64(r * Float64(1.0 / fma(b, Float64(b * -0.5), 1.0))))
end

Wolfram

code[r_, a_, b_] := N[(b * N[(r * N[(1.0 / N[(b * N[(b * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.1%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. --lowering--.f64 N/A

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

   7. +-rgt-identity N/A

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(b, \color{blue}{\left(b \cdot \left(\frac{-1}{2} \cdot \cos a\right) + 0\right)} - \sin a, \cos a\right)} \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{-1}{2} \cdot \cos a, 0\right)} - \sin a, \cos a\right)} \]

   9. *-commutative N/A

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\cos a \cdot \frac{-1}{2}}, 0\right) - \sin a, \cos a\right)} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\cos a \cdot \frac{-1}{2}}, 0\right) - \sin a, \cos a\right)} \]

   11. cos-lowering-cos.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\cos a} \cdot \frac{-1}{2}, 0\right) - \sin a, \cos a\right)} \]

   12. sin-lowering-sin.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \cos a \cdot \frac{-1}{2}, 0\right) - \color{blue}{\sin a}, \cos a\right)} \]

   13. cos-lowering-cos.f64 52.5

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \cos a \cdot -0.5, 0\right) - \sin a, \color{blue}{\cos a}\right)} \]

5. Simplified 52.5%

   \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \cos a \cdot -0.5, 0\right) - \sin a, \cos a\right)}} \]

6. Taylor expanded in b around 0

   \[\leadsto \frac{r \cdot \color{blue}{b}}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \cos a \cdot \frac{-1}{2}, 0\right) - \sin a, \cos a\right)} \]
7. Step-by-step derivation

8. Simplified 52.0%

   \[\leadsto \frac{r \cdot \color{blue}{b}}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \cos a \cdot -0.5, 0\right) - \sin a, \cos a\right)} \]

9. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{b \cdot r}{1 + \frac{-1}{2} \cdot {b}^{2}}} \]
10. Step-by-step derivation

    1. associate-/l* N/A

       \[\leadsto \color{blue}{b \cdot \frac{r}{1 + \frac{-1}{2} \cdot {b}^{2}}} \]

    2. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{b \cdot \frac{r}{1 + \frac{-1}{2} \cdot {b}^{2}}} \]

    3. /-lowering-/.f64 N/A

       \[\leadsto b \cdot \color{blue}{\frac{r}{1 + \frac{-1}{2} \cdot {b}^{2}}} \]

    4. +-commutative N/A

       \[\leadsto b \cdot \frac{r}{\color{blue}{\frac{-1}{2} \cdot {b}^{2} + 1}} \]

    5. *-commutative N/A

       \[\leadsto b \cdot \frac{r}{\color{blue}{{b}^{2} \cdot \frac{-1}{2}} + 1} \]

    6. unpow2 N/A

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

    7. associate-*l* N/A

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

    8. *-commutative N/A

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

    9. accelerator-lowering-fma.f64 N/A

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

    10. *-commutative N/A

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

    11. *-lowering-*.f64 36.1

        \[\leadsto b \cdot \frac{r}{\mathsf{fma}\left(b, \color{blue}{b \cdot -0.5}, 1\right)} \]

11. Simplified 36.1%

    \[\leadsto \color{blue}{b \cdot \frac{r}{\mathsf{fma}\left(b, b \cdot -0.5, 1\right)}} \]
12. Step-by-step derivation

    1. clear-num N/A

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

    2. associate-/r/ N/A

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

    3. *-lowering-*.f64 N/A

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

    4. /-lowering-/.f64 N/A

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

    5. accelerator-lowering-fma.f64 N/A

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

    6. *-lowering-*.f64 36.1

       \[\leadsto b \cdot \left(\frac{1}{\mathsf{fma}\left(b, \color{blue}{b \cdot -0.5}, 1\right)} \cdot r\right) \]

13. Applied egg-rr 36.1%

    \[\leadsto b \cdot \color{blue}{\left(\frac{1}{\mathsf{fma}\left(b, b \cdot -0.5, 1\right)} \cdot r\right)} \]

14. Final simplification 36.1%

    \[\leadsto b \cdot \left(r \cdot \frac{1}{\mathsf{fma}\left(b, b \cdot -0.5, 1\right)}\right) \]
15. Add Preprocessing

Alternative 14: 35.4% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ b \cdot \frac{r}{\mathsf{fma}\left(b, b \cdot -0.5, 1\right)} \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* b (/ r (fma b (* b -0.5) 1.0))))

C

double code(double r, double a, double b) {
	return b * (r / fma(b, (b * -0.5), 1.0));
}

Julia

function code(r, a, b)
	return Float64(b * Float64(r / fma(b, Float64(b * -0.5), 1.0)))
end

Wolfram

code[r_, a_, b_] := N[(b * N[(r / N[(b * N[(b * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.1%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. --lowering--.f64 N/A

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

   7. +-rgt-identity N/A

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(b, \color{blue}{\left(b \cdot \left(\frac{-1}{2} \cdot \cos a\right) + 0\right)} - \sin a, \cos a\right)} \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(b, \color{blue}{\mathsf{fma}\left(b, \frac{-1}{2} \cdot \cos a, 0\right)} - \sin a, \cos a\right)} \]

   9. *-commutative N/A

      \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\cos a \cdot \frac{-1}{2}}, 0\right) - \sin a, \cos a\right)} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\cos a \cdot \frac{-1}{2}}, 0\right) - \sin a, \cos a\right)} \]

   11. cos-lowering-cos.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \color{blue}{\cos a} \cdot \frac{-1}{2}, 0\right) - \sin a, \cos a\right)} \]

   12. sin-lowering-sin.f64 N/A

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \cos a \cdot \frac{-1}{2}, 0\right) - \color{blue}{\sin a}, \cos a\right)} \]

   13. cos-lowering-cos.f64 52.5

       \[\leadsto \frac{r \cdot \sin b}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \cos a \cdot -0.5, 0\right) - \sin a, \color{blue}{\cos a}\right)} \]

5. Simplified 52.5%

   \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \cos a \cdot -0.5, 0\right) - \sin a, \cos a\right)}} \]

6. Taylor expanded in b around 0

   \[\leadsto \frac{r \cdot \color{blue}{b}}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \cos a \cdot \frac{-1}{2}, 0\right) - \sin a, \cos a\right)} \]
7. Step-by-step derivation

8. Simplified 52.0%

   \[\leadsto \frac{r \cdot \color{blue}{b}}{\mathsf{fma}\left(b, \mathsf{fma}\left(b, \cos a \cdot -0.5, 0\right) - \sin a, \cos a\right)} \]

9. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{b \cdot r}{1 + \frac{-1}{2} \cdot {b}^{2}}} \]
10. Step-by-step derivation

    1. associate-/l* N/A

       \[\leadsto \color{blue}{b \cdot \frac{r}{1 + \frac{-1}{2} \cdot {b}^{2}}} \]

    2. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{b \cdot \frac{r}{1 + \frac{-1}{2} \cdot {b}^{2}}} \]

    3. /-lowering-/.f64 N/A

       \[\leadsto b \cdot \color{blue}{\frac{r}{1 + \frac{-1}{2} \cdot {b}^{2}}} \]

    4. +-commutative N/A

       \[\leadsto b \cdot \frac{r}{\color{blue}{\frac{-1}{2} \cdot {b}^{2} + 1}} \]

    5. *-commutative N/A

       \[\leadsto b \cdot \frac{r}{\color{blue}{{b}^{2} \cdot \frac{-1}{2}} + 1} \]

    6. unpow2 N/A

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

    7. associate-*l* N/A

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

    8. *-commutative N/A

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

    9. accelerator-lowering-fma.f64 N/A

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

    10. *-commutative N/A

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

    11. *-lowering-*.f64 36.1

        \[\leadsto b \cdot \frac{r}{\mathsf{fma}\left(b, \color{blue}{b \cdot -0.5}, 1\right)} \]

11. Simplified 36.1%

    \[\leadsto \color{blue}{b \cdot \frac{r}{\mathsf{fma}\left(b, b \cdot -0.5, 1\right)}} \]
12. Add Preprocessing

Alternative 15: 35.0% 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 74.1%

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

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

   4. cos-lowering-cos.f64 51.2

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

5. Simplified 51.2%

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

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. *-lowering-*.f64 36.0

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

8. Simplified 36.0%

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

Reproduce

herbie shell --seed 2024199 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))