rsin A (should all be same)

Percentage Accurate: 76.1% → 99.5%

Time: 13.3s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. cos-sum N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

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

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

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

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

   7. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

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

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

   13. cos-lowering-cos.f64 99.4

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

4. Applied egg-rr 99.4%

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

   1. flip3-- N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\frac{{0}^{3} - {\sin b}^{3}}{0 \cdot 0 + \left(\sin b \cdot \sin b + 0 \cdot \sin b\right)}} \cdot \sin a + \cos a \cdot \cos b} \]

   2. associate-*l/ N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\frac{\left({0}^{3} - {\sin b}^{3}\right) \cdot \sin a}{0 \cdot 0 + \left(\sin b \cdot \sin b + 0 \cdot \sin b\right)}} + \cos a \cdot \cos b} \]

   3. metadata-eval N/A

      \[\leadsto \frac{r \cdot \sin b}{\frac{\left(\color{blue}{0} - {\sin b}^{3}\right) \cdot \sin a}{0 \cdot 0 + \left(\sin b \cdot \sin b + 0 \cdot \sin b\right)} + \cos a \cdot \cos b} \]

   4. sub0-neg N/A

      \[\leadsto \frac{r \cdot \sin b}{\frac{\color{blue}{\left(\mathsf{neg}\left({\sin b}^{3}\right)\right)} \cdot \sin a}{0 \cdot 0 + \left(\sin b \cdot \sin b + 0 \cdot \sin b\right)} + \cos a \cdot \cos b} \]

   5. cube-neg N/A

      \[\leadsto \frac{r \cdot \sin b}{\frac{\color{blue}{{\left(\mathsf{neg}\left(\sin b\right)\right)}^{3}} \cdot \sin a}{0 \cdot 0 + \left(\sin b \cdot \sin b + 0 \cdot \sin b\right)} + \cos a \cdot \cos b} \]

   6. metadata-eval N/A

      \[\leadsto \frac{r \cdot \sin b}{\frac{{\left(\mathsf{neg}\left(\sin b\right)\right)}^{3} \cdot \sin a}{\color{blue}{0} + \left(\sin b \cdot \sin b + 0 \cdot \sin b\right)} + \cos a \cdot \cos b} \]

   7. +-lft-identity N/A

      \[\leadsto \frac{r \cdot \sin b}{\frac{{\left(\mathsf{neg}\left(\sin b\right)\right)}^{3} \cdot \sin a}{\color{blue}{\sin b \cdot \sin b + 0 \cdot \sin b}} + \cos a \cdot \cos b} \]

   8. distribute-rgt-out N/A

      \[\leadsto \frac{r \cdot \sin b}{\frac{{\left(\mathsf{neg}\left(\sin b\right)\right)}^{3} \cdot \sin a}{\color{blue}{\sin b \cdot \left(\sin b + 0\right)}} + \cos a \cdot \cos b} \]

   9. +-commutative N/A

      \[\leadsto \frac{r \cdot \sin b}{\frac{{\left(\mathsf{neg}\left(\sin b\right)\right)}^{3} \cdot \sin a}{\sin b \cdot \color{blue}{\left(0 + \sin b\right)}} + \cos a \cdot \cos b} \]

   10. +-lft-identity N/A

       \[\leadsto \frac{r \cdot \sin b}{\frac{{\left(\mathsf{neg}\left(\sin b\right)\right)}^{3} \cdot \sin a}{\sin b \cdot \color{blue}{\sin b}} + \cos a \cdot \cos b} \]

   11. times-frac N/A

       \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\frac{{\left(\mathsf{neg}\left(\sin b\right)\right)}^{3}}{\sin b} \cdot \frac{\sin a}{\sin b}} + \cos a \cdot \cos b} \]

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

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

6. Applied egg-rr 99.4%

   \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\frac{0 - {\sin b}^{3}}{\sin b}, \frac{\sin a}{\sin b}, \cos a \cdot \cos b\right)}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{\frac{0 - {\sin b}^{3}}{\sin b} \cdot \frac{\sin a}{\sin b} + \cos a \cdot \cos b} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{\sin b \cdot \frac{r}{\frac{0 - {\sin b}^{3}}{\sin b} \cdot \frac{\sin a}{\sin b} + \cos a \cdot \cos b}} \]

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

      \[\leadsto \color{blue}{\sin b \cdot \frac{r}{\frac{0 - {\sin b}^{3}}{\sin b} \cdot \frac{\sin a}{\sin b} + \cos a \cdot \cos b}} \]

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

      \[\leadsto \color{blue}{\sin b} \cdot \frac{r}{\frac{0 - {\sin b}^{3}}{\sin b} \cdot \frac{\sin a}{\sin b} + \cos a \cdot \cos b} \]

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

      \[\leadsto \sin b \cdot \color{blue}{\frac{r}{\frac{0 - {\sin b}^{3}}{\sin b} \cdot \frac{\sin a}{\sin b} + \cos a \cdot \cos b}} \]

8. Applied egg-rr 99.5%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. cos-sum N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

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

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

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

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

   7. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

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

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

   13. cos-lowering-cos.f64 99.4

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

4. Applied egg-rr 99.4%

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

   1. sub0-neg N/A

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

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

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

   3. sin-lowering-sin.f64 99.4

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. cos-sum N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

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

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

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

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

   7. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

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

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

   13. cos-lowering-cos.f64 99.4

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

4. Applied egg-rr 99.4%

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

   1. flip3-- N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\frac{{0}^{3} - {\sin b}^{3}}{0 \cdot 0 + \left(\sin b \cdot \sin b + 0 \cdot \sin b\right)}} \cdot \sin a + \cos a \cdot \cos b} \]

   2. associate-*l/ N/A

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\frac{\left({0}^{3} - {\sin b}^{3}\right) \cdot \sin a}{0 \cdot 0 + \left(\sin b \cdot \sin b + 0 \cdot \sin b\right)}} + \cos a \cdot \cos b} \]

   3. metadata-eval N/A

      \[\leadsto \frac{r \cdot \sin b}{\frac{\left(\color{blue}{0} - {\sin b}^{3}\right) \cdot \sin a}{0 \cdot 0 + \left(\sin b \cdot \sin b + 0 \cdot \sin b\right)} + \cos a \cdot \cos b} \]

   4. sub0-neg N/A

      \[\leadsto \frac{r \cdot \sin b}{\frac{\color{blue}{\left(\mathsf{neg}\left({\sin b}^{3}\right)\right)} \cdot \sin a}{0 \cdot 0 + \left(\sin b \cdot \sin b + 0 \cdot \sin b\right)} + \cos a \cdot \cos b} \]

   5. cube-neg N/A

      \[\leadsto \frac{r \cdot \sin b}{\frac{\color{blue}{{\left(\mathsf{neg}\left(\sin b\right)\right)}^{3}} \cdot \sin a}{0 \cdot 0 + \left(\sin b \cdot \sin b + 0 \cdot \sin b\right)} + \cos a \cdot \cos b} \]

   6. metadata-eval N/A

      \[\leadsto \frac{r \cdot \sin b}{\frac{{\left(\mathsf{neg}\left(\sin b\right)\right)}^{3} \cdot \sin a}{\color{blue}{0} + \left(\sin b \cdot \sin b + 0 \cdot \sin b\right)} + \cos a \cdot \cos b} \]

   7. +-lft-identity N/A

      \[\leadsto \frac{r \cdot \sin b}{\frac{{\left(\mathsf{neg}\left(\sin b\right)\right)}^{3} \cdot \sin a}{\color{blue}{\sin b \cdot \sin b + 0 \cdot \sin b}} + \cos a \cdot \cos b} \]

   8. distribute-rgt-out N/A

      \[\leadsto \frac{r \cdot \sin b}{\frac{{\left(\mathsf{neg}\left(\sin b\right)\right)}^{3} \cdot \sin a}{\color{blue}{\sin b \cdot \left(\sin b + 0\right)}} + \cos a \cdot \cos b} \]

   9. +-commutative N/A

      \[\leadsto \frac{r \cdot \sin b}{\frac{{\left(\mathsf{neg}\left(\sin b\right)\right)}^{3} \cdot \sin a}{\sin b \cdot \color{blue}{\left(0 + \sin b\right)}} + \cos a \cdot \cos b} \]

   10. +-lft-identity N/A

       \[\leadsto \frac{r \cdot \sin b}{\frac{{\left(\mathsf{neg}\left(\sin b\right)\right)}^{3} \cdot \sin a}{\sin b \cdot \color{blue}{\sin b}} + \cos a \cdot \cos b} \]

   11. times-frac N/A

       \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\frac{{\left(\mathsf{neg}\left(\sin b\right)\right)}^{3}}{\sin b} \cdot \frac{\sin a}{\sin b}} + \cos a \cdot \cos b} \]

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

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

6. Applied egg-rr 99.4%

   \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\mathsf{fma}\left(\frac{0 - {\sin b}^{3}}{\sin b}, \frac{\sin a}{\sin b}, \cos a \cdot \cos b\right)}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{\frac{0 - {\sin b}^{3}}{\sin b} \cdot \frac{\sin a}{\sin b} + \cos a \cdot \cos b} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{\sin b \cdot \frac{r}{\frac{0 - {\sin b}^{3}}{\sin b} \cdot \frac{\sin a}{\sin b} + \cos a \cdot \cos b}} \]

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

      \[\leadsto \color{blue}{\sin b \cdot \frac{r}{\frac{0 - {\sin b}^{3}}{\sin b} \cdot \frac{\sin a}{\sin b} + \cos a \cdot \cos b}} \]

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

      \[\leadsto \color{blue}{\sin b} \cdot \frac{r}{\frac{0 - {\sin b}^{3}}{\sin b} \cdot \frac{\sin a}{\sin b} + \cos a \cdot \cos b} \]

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

      \[\leadsto \sin b \cdot \color{blue}{\frac{r}{\frac{0 - {\sin b}^{3}}{\sin b} \cdot \frac{\sin a}{\sin b} + \cos a \cdot \cos b}} \]

8. Applied egg-rr 99.5%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. sub0-neg N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. sin-lowering-sin.f64 99.4

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

10. Applied egg-rr 99.4%

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

Alternative 4: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

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

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

   7. +-commutative N/A

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

   8. +-lowering-+.f64 79.5

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

4. Applied egg-rr 79.5%

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

5. Final simplification 79.5%

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

Alternative 5: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. +-commutative N/A

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

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

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

   9. sin-lowering-sin.f64 79.5

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

4. Applied egg-rr 79.5%

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

5. Final simplification 79.5%

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

Alternative 6: 75.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-10}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (tan b))))
   (if (<= b -1.32e-5) t_0 (if (<= b 1.35e-10) (* r (/ b (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * tan(b);
	double tmp;
	if (b <= -1.32e-5) {
		tmp = t_0;
	} else if (b <= 1.35e-10) {
		tmp = r * (b / cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * tan(b)
    if (b <= (-1.32d-5)) then
        tmp = t_0
    else if (b <= 1.35d-10) then
        tmp = r * (b / cos(a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.tan(b);
	double tmp;
	if (b <= -1.32e-5) {
		tmp = t_0;
	} else if (b <= 1.35e-10) {
		tmp = r * (b / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.tan(b)
	tmp = 0
	if b <= -1.32e-5:
		tmp = t_0
	elif b <= 1.35e-10:
		tmp = r * (b / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * tan(b))
	tmp = 0.0
	if (b <= -1.32e-5)
		tmp = t_0;
	elseif (b <= 1.35e-10)
		tmp = Float64(r * Float64(b / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * tan(b);
	tmp = 0.0;
	if (b <= -1.32e-5)
		tmp = t_0;
	elseif (b <= 1.35e-10)
		tmp = r * (b / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.32e-5], t$95$0, If[LessEqual[b, 1.35e-10], N[(r * N[(b / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.32000000000000007e-5 or 1.35e-10 < b

   1. Initial program 63.3%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

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

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

      4. cos-lowering-cos.f64 62.8

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

   5. Simplified 62.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. quot-tan N/A

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

      5. tan-lowering-tan.f64 63.0

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

   7. Applied egg-rr 63.0%

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

   if -1.32000000000000007e-5 < b < 1.35e-10

   1. Initial program 99.5%

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

   3. Taylor expanded in b around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. cos-lowering-cos.f64 99.5

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

   5. Simplified 99.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. cos-lowering-cos.f64 99.6

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

   7. Applied egg-rr 99.6%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.32 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-10}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-10}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (tan b))))
   (if (<= b -2.5e-5) t_0 (if (<= b 1.35e-10) (* b (/ r (cos a))) t_0))))

C

double code(double r, double a, double b) {
	double t_0 = r * tan(b);
	double tmp;
	if (b <= -2.5e-5) {
		tmp = t_0;
	} else if (b <= 1.35e-10) {
		tmp = b * (r / cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r * tan(b)
    if (b <= (-2.5d-5)) then
        tmp = t_0
    else if (b <= 1.35d-10) then
        tmp = b * (r / cos(a))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double t_0 = r * Math.tan(b);
	double tmp;
	if (b <= -2.5e-5) {
		tmp = t_0;
	} else if (b <= 1.35e-10) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(r, a, b):
	t_0 = r * math.tan(b)
	tmp = 0
	if b <= -2.5e-5:
		tmp = t_0
	elif b <= 1.35e-10:
		tmp = b * (r / math.cos(a))
	else:
		tmp = t_0
	return tmp

Julia

function code(r, a, b)
	t_0 = Float64(r * tan(b))
	tmp = 0.0
	if (b <= -2.5e-5)
		tmp = t_0;
	elseif (b <= 1.35e-10)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	t_0 = r * tan(b);
	tmp = 0.0;
	if (b <= -2.5e-5)
		tmp = t_0;
	elseif (b <= 1.35e-10)
		tmp = b * (r / cos(a));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := Block[{t$95$0 = N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.5e-5], t$95$0, If[LessEqual[b, 1.35e-10], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -2.50000000000000012e-5 or 1.35e-10 < b

   1. Initial program 63.3%

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

   3. Taylor expanded in a around 0

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

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

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

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

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

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

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

      4. cos-lowering-cos.f64 62.8

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

   5. Simplified 62.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. quot-tan N/A

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

      5. tan-lowering-tan.f64 63.0

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

   7. Applied egg-rr 63.0%

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

   if -2.50000000000000012e-5 < b < 1.35e-10

   1. Initial program 99.5%

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

   3. Taylor expanded in b around 0

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

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

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

      2. *-commutative N/A

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

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

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

      4. cos-lowering-cos.f64 99.5

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

   5. Simplified 99.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. cos-lowering-cos.f64 99.5

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

   7. Applied egg-rr 99.5%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-10}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

3. Taylor expanded in a around 0

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

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

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

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

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

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

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

   4. cos-lowering-cos.f64 60.7

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

5. Simplified 60.7%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

   4. quot-tan N/A

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

   5. tan-lowering-tan.f64 60.8

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

7. Applied egg-rr 60.8%

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

8. Final simplification 60.8%

   \[\leadsto r \cdot \tan b \]
9. Add Preprocessing

Alternative 9: 38.5% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

3. Taylor expanded in a around 0

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

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

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

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

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

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

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

   4. cos-lowering-cos.f64 60.7

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

5. Simplified 60.7%

   \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos b}} \]
6. 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. *-lowering-*.f64 N/A

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

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

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

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

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

   6. cos-lowering-cos.f64 60.7

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

7. Applied egg-rr 60.7%

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

8. Taylor expanded in b around 0

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

10. Simplified 32.5%

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

Alternative 10: 34.4% accurate, 36.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

3. Taylor expanded in b around 0

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

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

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

   2. *-commutative N/A

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

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

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

   4. cos-lowering-cos.f64 47.1

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

5. Simplified 47.1%

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

6. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 28.7

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

8. Simplified 28.7%

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

9. Final simplification 28.7%

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

Reproduce

herbie shell --seed 2024195 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))