rsin B (should all be same)

Percentage Accurate: 77.0% → 99.5%

Time: 12.1s

Alternatives: 17

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in a around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. mul-1-neg N/A

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

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

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

   10. mul-1-neg N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-sin.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 52.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(a + b\right)\\ \mathbf{if}\;\frac{\sin b}{t\_0} \leq -0.05:\\ \;\;\;\;\frac{\sin b \cdot r}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{fma}\left(b \cdot b, 0.16666666666666666, 1\right)}{b}}}{t\_0} \cdot r\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (cos (+ a b))))
   (if (<= (/ (sin b) t_0) -0.05)
     (/ (* (sin b) r) 1.0)
     (* (/ (/ 1.0 (/ (fma (* b b) 0.16666666666666666 1.0) b)) t_0) r))))

C

double code(double r, double a, double b) {
	double t_0 = cos((a + b));
	double tmp;
	if ((sin(b) / t_0) <= -0.05) {
		tmp = (sin(b) * r) / 1.0;
	} else {
		tmp = ((1.0 / (fma((b * b), 0.16666666666666666, 1.0) / b)) / t_0) * r;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = cos(Float64(a + b))
	tmp = 0.0
	if (Float64(sin(b) / t_0) <= -0.05)
		tmp = Float64(Float64(sin(b) * r) / 1.0);
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(fma(Float64(b * b), 0.16666666666666666, 1.0) / b)) / t_0) * r);
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Sin[b], $MachinePrecision] / t$95$0), $MachinePrecision], -0.05], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / 1.0), $MachinePrecision], N[(N[(N[(1.0 / N[(N[(N[(b * b), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * r), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b))) < -0.050000000000000003

   1. Initial program 48.3%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-sin.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sin.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 47.9%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 13.5%

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

   if -0.050000000000000003 < (/.f64 (sin.f64 b) (cos.f64 (+.f64 a b)))

   1. Initial program 85.4%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 69.8

         \[\leadsto r \cdot \frac{\mathsf{fma}\left(\color{blue}{{b}^{3}}, -0.16666666666666666, b\right)}{\cos \left(a + b\right)} \]

   5. Applied rewrites 69.8%

      \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left({b}^{3}, -0.16666666666666666, b\right)}}{\cos \left(a + b\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.7%

      \[\leadsto r \cdot \frac{\frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left(-0.16666666666666666, {b}^{3}, b\right)}}}}{\cos \left(a + b\right)} \]

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 70.2%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin b}{\cos \left(a + b\right)} \leq -0.05:\\ \;\;\;\;\frac{\sin b \cdot r}{1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{fma}\left(b \cdot b, 0.16666666666666666, 1\right)}{b}}}{\cos \left(a + b\right)} \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin b \cdot r\\ \mathbf{if}\;a \leq -0.000102:\\ \;\;\;\;\frac{t\_0}{\cos a}\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_0}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) r)))
   (if (<= a -0.000102)
     (/ t_0 (cos a))
     (if (<= a 1.05e-5) (/ t_0 (cos b)) (* (/ (sin b) (cos a)) r)))))

C

double code(double r, double a, double b) {
	double t_0 = sin(b) * r;
	double tmp;
	if (a <= -0.000102) {
		tmp = t_0 / cos(a);
	} else if (a <= 1.05e-5) {
		tmp = t_0 / cos(b);
	} else {
		tmp = (sin(b) / cos(a)) * r;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(b) * r
    if (a <= (-0.000102d0)) then
        tmp = t_0 / cos(a)
    else if (a <= 1.05d-5) then
        tmp = t_0 / cos(b)
    else
        tmp = (sin(b) / cos(a)) * r
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = Math.sin(b) * r;
	double tmp;
	if (a <= -0.000102) {
		tmp = t_0 / Math.cos(a);
	} else if (a <= 1.05e-5) {
		tmp = t_0 / Math.cos(b);
	} else {
		tmp = (Math.sin(b) / Math.cos(a)) * r;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = math.sin(b) * r
	tmp = 0
	if a <= -0.000102:
		tmp = t_0 / math.cos(a)
	elif a <= 1.05e-5:
		tmp = t_0 / math.cos(b)
	else:
		tmp = (math.sin(b) / math.cos(a)) * r
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(sin(b) * r)
	tmp = 0.0
	if (a <= -0.000102)
		tmp = Float64(t_0 / cos(a));
	elseif (a <= 1.05e-5)
		tmp = Float64(t_0 / cos(b));
	else
		tmp = Float64(Float64(sin(b) / cos(a)) * r);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = sin(b) * r;
	tmp = 0.0;
	if (a <= -0.000102)
		tmp = t_0 / cos(a);
	elseif (a <= 1.05e-5)
		tmp = t_0 / cos(b);
	else
		tmp = (sin(b) / cos(a)) * r;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[a, -0.000102], N[(t$95$0 / N[Cos[a], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.05e-5], N[(t$95$0 / N[Cos[b], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.01999999999999999e-4

   1. Initial program 51.3%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-sin.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 51.6

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

   7. Applied rewrites 51.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 51.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 51.7

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

   9. Applied rewrites 51.7%

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

   if -1.01999999999999999e-4 < a < 1.04999999999999994e-5

   1. Initial program 99.3%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sin.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 99.3%

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

   if 1.04999999999999994e-5 < a

   1. Initial program 60.8%

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

   3. Taylor expanded in b around 0

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

      1. lower-cos.f64 60.8

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

   5. Applied rewrites 60.8%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.000102:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos a}\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.000102:\\ \;\;\;\;\frac{r}{\cos a} \cdot \sin b\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= a -0.000102)
   (* (/ r (cos a)) (sin b))
   (if (<= a 1.05e-5) (/ (* (sin b) r) (cos b)) (* (/ (sin b) (cos a)) r))))

C

double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.000102) {
		tmp = (r / cos(a)) * sin(b);
	} else if (a <= 1.05e-5) {
		tmp = (sin(b) * r) / cos(b);
	} else {
		tmp = (sin(b) / cos(a)) * r;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-0.000102d0)) then
        tmp = (r / cos(a)) * sin(b)
    else if (a <= 1.05d-5) then
        tmp = (sin(b) * r) / cos(b)
    else
        tmp = (sin(b) / cos(a)) * r
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (a <= -0.000102) {
		tmp = (r / Math.cos(a)) * Math.sin(b);
	} else if (a <= 1.05e-5) {
		tmp = (Math.sin(b) * r) / Math.cos(b);
	} else {
		tmp = (Math.sin(b) / Math.cos(a)) * r;
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if a <= -0.000102:
		tmp = (r / math.cos(a)) * math.sin(b)
	elif a <= 1.05e-5:
		tmp = (math.sin(b) * r) / math.cos(b)
	else:
		tmp = (math.sin(b) / math.cos(a)) * r
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (a <= -0.000102)
		tmp = Float64(Float64(r / cos(a)) * sin(b));
	elseif (a <= 1.05e-5)
		tmp = Float64(Float64(sin(b) * r) / cos(b));
	else
		tmp = Float64(Float64(sin(b) / cos(a)) * r);
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (a <= -0.000102)
		tmp = (r / cos(a)) * sin(b);
	elseif (a <= 1.05e-5)
		tmp = (sin(b) * r) / cos(b);
	else
		tmp = (sin(b) / cos(a)) * r;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[a, -0.000102], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.05e-5], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.01999999999999999e-4

   1. Initial program 51.3%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-sin.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in b around 0

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

      1. lower-cos.f64 51.6

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

   7. Applied rewrites 51.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 51.6

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

   9. Applied rewrites 51.6%

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

   if -1.01999999999999999e-4 < a < 1.04999999999999994e-5

   1. Initial program 99.3%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sin.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 99.3%

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

   if 1.04999999999999994e-5 < a

   1. Initial program 60.8%

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

   3. Taylor expanded in b around 0

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

      1. lower-cos.f64 60.8

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

   5. Applied rewrites 60.8%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.000102:\\ \;\;\;\;\frac{r}{\cos a} \cdot \sin b\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b}{\cos a} \cdot r\\ \mathbf{if}\;a \leq -0.000102:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ (sin b) (cos a)) r)))
   (if (<= a -0.000102)
     t_0
     (if (<= a 1.05e-5) (/ (* (sin b) r) (cos b)) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (sin(b) / cos(a)) * r;
	double tmp;
	if (a <= -0.000102) {
		tmp = t_0;
	} else if (a <= 1.05e-5) {
		tmp = (sin(b) * r) / cos(b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(b) / cos(a)) * r
    if (a <= (-0.000102d0)) then
        tmp = t_0
    else if (a <= 1.05d-5) then
        tmp = (sin(b) * r) / cos(b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (Math.sin(b) / Math.cos(a)) * r;
	double tmp;
	if (a <= -0.000102) {
		tmp = t_0;
	} else if (a <= 1.05e-5) {
		tmp = (Math.sin(b) * r) / Math.cos(b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (math.sin(b) / math.cos(a)) * r
	tmp = 0
	if a <= -0.000102:
		tmp = t_0
	elif a <= 1.05e-5:
		tmp = (math.sin(b) * r) / math.cos(b)
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(sin(b) / cos(a)) * r)
	tmp = 0.0
	if (a <= -0.000102)
		tmp = t_0;
	elseif (a <= 1.05e-5)
		tmp = Float64(Float64(sin(b) * r) / cos(b));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (sin(b) / cos(a)) * r;
	tmp = 0.0;
	if (a <= -0.000102)
		tmp = t_0;
	elseif (a <= 1.05e-5)
		tmp = (sin(b) * r) / cos(b);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision]}, If[LessEqual[a, -0.000102], t$95$0, If[LessEqual[a, 1.05e-5], N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.01999999999999999e-4 or 1.04999999999999994e-5 < a

   1. Initial program 56.0%

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

   3. Taylor expanded in b around 0

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

      1. lower-cos.f64 56.2

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

   5. Applied rewrites 56.2%

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

   if -1.01999999999999999e-4 < a < 1.04999999999999994e-5

   1. Initial program 99.3%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sin.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 99.3%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.000102:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\cos a} \cdot r\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b \cdot r}{\cos b}\\ \mathbf{if}\;b \leq -3.2 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-7}:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (* (sin b) r) (cos b))))
   (if (<= b -3.2e-5) t_0 (if (<= b 3e-7) (* (/ r (cos a)) b) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (sin(b) * r) / cos(b);
	double tmp;
	if (b <= -3.2e-5) {
		tmp = t_0;
	} else if (b <= 3e-7) {
		tmp = (r / cos(a)) * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(b) * r) / cos(b)
    if (b <= (-3.2d-5)) then
        tmp = t_0
    else if (b <= 3d-7) then
        tmp = (r / cos(a)) * b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (Math.sin(b) * r) / Math.cos(b);
	double tmp;
	if (b <= -3.2e-5) {
		tmp = t_0;
	} else if (b <= 3e-7) {
		tmp = (r / Math.cos(a)) * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (math.sin(b) * r) / math.cos(b)
	tmp = 0
	if b <= -3.2e-5:
		tmp = t_0
	elif b <= 3e-7:
		tmp = (r / math.cos(a)) * b
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(sin(b) * r) / cos(b))
	tmp = 0.0
	if (b <= -3.2e-5)
		tmp = t_0;
	elseif (b <= 3e-7)
		tmp = Float64(Float64(r / cos(a)) * b);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (sin(b) * r) / cos(b);
	tmp = 0.0;
	if (b <= -3.2e-5)
		tmp = t_0;
	elseif (b <= 3e-7)
		tmp = (r / cos(a)) * b;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.2e-5], t$95$0, If[LessEqual[b, 3e-7], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.19999999999999986e-5 or 2.9999999999999999e-7 < b

   1. Initial program 51.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-sin.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sin.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 51.7%

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

   if -3.19999999999999986e-5 < b < 2.9999999999999999e-7

   1. Initial program 99.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 99.2

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

   7. Applied rewrites 99.2%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-7}:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{r}{\cos b} \cdot \sin b\\ \mathbf{if}\;b \leq -3.2 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-7}:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (/ r (cos b)) (sin b))))
   (if (<= b -3.2e-5) t_0 (if (<= b 3e-7) (* (/ r (cos a)) b) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (r / cos(b)) * sin(b);
	double tmp;
	if (b <= -3.2e-5) {
		tmp = t_0;
	} else if (b <= 3e-7) {
		tmp = (r / cos(a)) * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (r / cos(b)) * sin(b)
    if (b <= (-3.2d-5)) then
        tmp = t_0
    else if (b <= 3d-7) then
        tmp = (r / cos(a)) * b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (r / Math.cos(b)) * Math.sin(b);
	double tmp;
	if (b <= -3.2e-5) {
		tmp = t_0;
	} else if (b <= 3e-7) {
		tmp = (r / Math.cos(a)) * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (r / math.cos(b)) * math.sin(b)
	tmp = 0
	if b <= -3.2e-5:
		tmp = t_0
	elif b <= 3e-7:
		tmp = (r / math.cos(a)) * b
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(r / cos(b)) * sin(b))
	tmp = 0.0
	if (b <= -3.2e-5)
		tmp = t_0;
	elseif (b <= 3e-7)
		tmp = Float64(Float64(r / cos(a)) * b);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (r / cos(b)) * sin(b);
	tmp = 0.0;
	if (b <= -3.2e-5)
		tmp = t_0;
	elseif (b <= 3e-7)
		tmp = (r / cos(a)) * b;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(r / N[Cos[b], $MachinePrecision]), $MachinePrecision] * N[Sin[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.2e-5], t$95$0, If[LessEqual[b, 3e-7], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.19999999999999986e-5 or 2.9999999999999999e-7 < b

   1. Initial program 51.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 51.6

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

   5. Applied rewrites 51.6%

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

   if -3.19999999999999986e-5 < b < 2.9999999999999999e-7

   1. Initial program 99.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 99.2

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

   7. Applied rewrites 99.2%

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

Alternative 8: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 76.3

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

4. Applied rewrites 76.3%

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

Alternative 9: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. cube-unmult N/A

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

   9. lower-pow.f64 53.2

      \[\leadsto r \cdot \frac{\mathsf{fma}\left(\color{blue}{{b}^{3}}, -0.16666666666666666, b\right)}{\cos \left(a + b\right)} \]

5. Applied rewrites 53.2%

   \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left({b}^{3}, -0.16666666666666666, b\right)}}{\cos \left(a + b\right)} \]
6. Step-by-step derivation

7. Applied rewrites 53.1%

   \[\leadsto r \cdot \frac{\frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left(-0.16666666666666666, {b}^{3}, b\right)}}}}{\cos \left(a + b\right)} \]

8. Taylor expanded in b around inf

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

   1. lower-sin.f64 76.3

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

10. Applied rewrites 76.3%

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

11. Final simplification 76.3%

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

Alternative 10: 54.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b \cdot r}{1}\\ \mathbf{if}\;b \leq -2.8:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{+33}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666 \cdot \left(b \cdot b\right), b, b\right)}{\cos \left(a + b\right)} \cdot r\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (* (sin b) r) 1.0)))
   (if (<= b -2.8)
     t_0
     (if (<= b 1.4e+33)
       (* (/ (fma (* -0.16666666666666666 (* b b)) b b) (cos (+ a b))) r)
       t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (sin(b) * r) / 1.0;
	double tmp;
	if (b <= -2.8) {
		tmp = t_0;
	} else if (b <= 1.4e+33) {
		tmp = (fma((-0.16666666666666666 * (b * b)), b, b) / cos((a + b))) * r;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(r, a, b)
	t_0 = Float64(Float64(sin(b) * r) / 1.0)
	tmp = 0.0
	if (b <= -2.8)
		tmp = t_0;
	elseif (b <= 1.4e+33)
		tmp = Float64(Float64(fma(Float64(-0.16666666666666666 * Float64(b * b)), b, b) / cos(Float64(a + b))) * r);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / 1.0), $MachinePrecision]}, If[LessEqual[b, -2.8], t$95$0, If[LessEqual[b, 1.4e+33], N[(N[(N[(N[(-0.16666666666666666 * N[(b * b), $MachinePrecision]), $MachinePrecision] * b + b), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * r), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.7999999999999998 or 1.4e33 < b

   1. Initial program 51.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-sin.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sin.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 51.1%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 12.3%

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

   if -2.7999999999999998 < b < 1.4e33

   1. Initial program 98.5%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. cube-unmult N/A

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

      9. lower-pow.f64 98.5

         \[\leadsto r \cdot \frac{\mathsf{fma}\left(\color{blue}{{b}^{3}}, -0.16666666666666666, b\right)}{\cos \left(a + b\right)} \]

   5. Applied rewrites 98.5%

      \[\leadsto r \cdot \frac{\color{blue}{\mathsf{fma}\left({b}^{3}, -0.16666666666666666, b\right)}}{\cos \left(a + b\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.5%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8:\\ \;\;\;\;\frac{\sin b \cdot r}{1}\\ \mathbf{elif}\;b \leq 1.4 \cdot 10^{+33}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.16666666666666666 \cdot \left(b \cdot b\right), b, b\right)}{\cos \left(a + b\right)} \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin b \cdot r}{1}\\ \mathbf{if}\;b \leq -4.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{+34}:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (/ (* (sin b) r) 1.0)))
   (if (<= b -4.5) t_0 (if (<= b 3.2e+34) (* (/ r (cos a)) b) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = (sin(b) * r) / 1.0;
	double tmp;
	if (b <= -4.5) {
		tmp = t_0;
	} else if (b <= 3.2e+34) {
		tmp = (r / cos(a)) * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(b) * r) / 1.0d0
    if (b <= (-4.5d0)) then
        tmp = t_0
    else if (b <= 3.2d+34) then
        tmp = (r / cos(a)) * b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = (Math.sin(b) * r) / 1.0;
	double tmp;
	if (b <= -4.5) {
		tmp = t_0;
	} else if (b <= 3.2e+34) {
		tmp = (r / Math.cos(a)) * b;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = (math.sin(b) * r) / 1.0
	tmp = 0
	if b <= -4.5:
		tmp = t_0
	elif b <= 3.2e+34:
		tmp = (r / math.cos(a)) * b
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(Float64(sin(b) * r) / 1.0)
	tmp = 0.0
	if (b <= -4.5)
		tmp = t_0;
	elseif (b <= 3.2e+34)
		tmp = Float64(Float64(r / cos(a)) * b);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = (sin(b) * r) / 1.0;
	tmp = 0.0;
	if (b <= -4.5)
		tmp = t_0;
	elseif (b <= 3.2e+34)
		tmp = (r / cos(a)) * b;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / 1.0), $MachinePrecision]}, If[LessEqual[b, -4.5], t$95$0, If[LessEqual[b, 3.2e+34], N[(N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -4.5 or 3.1999999999999998e34 < b

   1. Initial program 51.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. cos-sum N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-sin.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. mul-1-neg N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-sin.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 51.3%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 12.4%

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

   if -4.5 < b < 3.1999999999999998e34

   1. Initial program 98.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. inv-pow N/A

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

      7. lower-pow.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-cos.f64 97.7

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

   7. Applied rewrites 97.7%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5:\\ \;\;\;\;\frac{\sin b \cdot r}{1}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{+34}:\\ \;\;\;\;\frac{r}{\cos a} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 38.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\sin b \cdot r}{1} \end{array} \]

FPCore

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

C

double code(double r, double a, double b) {
	return (sin(b) * r) / 1.0;
}

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) / 1.0d0
end function

Java

public static double code(double r, double a, double b) {
	return (Math.sin(b) * r) / 1.0;
}

Python

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

Julia

function code(r, a, b)
	return Float64(Float64(sin(b) * r) / 1.0)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in a around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. mul-1-neg N/A

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

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

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

   10. mul-1-neg N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-sin.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 60.3%

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

11. Taylor expanded in b around 0

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

13. Applied rewrites 41.8%

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

14. Final simplification 41.8%

    \[\leadsto \frac{\sin b \cdot r}{1} \]
15. Add Preprocessing

Alternative 13: 34.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. cos-sum N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. *-commutative N/A

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

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

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

   12. lower-*.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in a around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. mul-1-neg N/A

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

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

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

   10. mul-1-neg N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-sin.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 60.3%

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

11. Taylor expanded in b around 0

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

13. Applied rewrites 37.9%

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

Alternative 14: 34.3% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \frac{b}{1} \cdot r \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (b / 1.0d0) * r
end function

Java

public static double code(double r, double a, double b) {
	return (b / 1.0) * r;
}

Python

def code(r, a, b):
	return (b / 1.0) * r

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

3. Taylor expanded in b around 0

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

   1. lower-/.f64 N/A

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

   2. lower-cos.f64 53.6

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

5. Applied rewrites 53.6%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 37.6%

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

9. Final simplification 37.6%

   \[\leadsto \frac{b}{1} \cdot r \]
10. Add Preprocessing

Alternative 15: 32.5% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. associate-*r/ N/A

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

   14. *-rgt-identity N/A

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

   15. lower-/.f64 N/A

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

   16. lower-cos.f64 53.4

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

5. Applied rewrites 53.4%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 35.3%

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

9. Taylor expanded in a around inf

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

11. Applied rewrites 16.5%

    \[\leadsto r \cdot \left(\left(b \cdot b\right) \cdot a\right) \]

12. Taylor expanded in a around 0

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

14. Applied rewrites 35.3%

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

15. Final simplification 35.3%

    \[\leadsto \mathsf{fma}\left(b \cdot b, a, b\right) \cdot r \]
16. Add Preprocessing

Alternative 16: 32.1% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. associate-*r/ N/A

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

   14. *-rgt-identity N/A

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

   15. lower-/.f64 N/A

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

   16. lower-cos.f64 53.4

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

5. Applied rewrites 53.4%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 35.3%

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

10. Applied rewrites 35.2%

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

11. Final simplification 35.2%

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

Alternative 17: 14.1% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. unpow2 N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. associate-*r/ N/A

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

   14. *-rgt-identity N/A

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

   15. lower-/.f64 N/A

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

   16. lower-cos.f64 53.4

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

5. Applied rewrites 53.4%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 35.3%

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

9. Taylor expanded in a around inf

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

11. Applied rewrites 16.5%

    \[\leadsto r \cdot \left(\left(b \cdot b\right) \cdot a\right) \]
12. Step-by-step derivation

13. Applied rewrites 16.6%

    \[\leadsto r \cdot \left(\left(a \cdot b\right) \cdot b\right) \]

14. Final simplification 16.6%

    \[\leadsto \left(\left(a \cdot b\right) \cdot b\right) \cdot r \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024273 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))