rsin A (should all be same)

Percentage Accurate: 76.3% → 99.5%

Time: 13.5s

Alternatives: 9

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: 76.3% 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, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

   6. +-lowering-+.f64 80.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

3. Simplified 80.2%

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

   1. cos-sum N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}\right)\right) \]

   2. fmm-def N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \mathsf{neg}\left(\sin b \cdot \sin a\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\cos b, \color{blue}{\cos a}, \left(\mathsf{neg}\left(\sin b \cdot \sin a\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \cos \color{blue}{a}, \left(\mathsf{neg}\left(\sin b \cdot \sin a\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \left(\mathsf{neg}\left(\sin b \cdot \sin a\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \left(\sin b \cdot \left(\mathsf{neg}\left(\sin a\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{*.f64}\left(\sin b, \left(\mathsf{neg}\left(\sin a\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \left(\mathsf{neg}\left(\sin a\right)\right)\right)\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \left(0 - \sin a\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{\_.f64}\left(0, \sin a\right)\right)\right)\right) \]

   11. sin-lowering-sin.f64 99.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(a\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.6%

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

   1. fma-define N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\mathsf{fma}\left(\cos b, \color{blue}{\cos a}, \sin b \cdot \left(0 - \sin a\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\cos b, \color{blue}{\cos a}, \left(\sin b \cdot \left(0 - \sin a\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \cos \color{blue}{a}, \left(\sin b \cdot \left(0 - \sin a\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \left(\sin b \cdot \left(0 - \sin a\right)\right)\right)\right) \]

   5. sub0-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \left(\sin b \cdot \left(\mathsf{neg}\left(\sin a\right)\right)\right)\right)\right) \]

   6. distribute-rgt-neg-out N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \left(\mathsf{neg}\left(\sin b \cdot \sin a\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{neg.f64}\left(\left(\sin b \cdot \sin a\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\sin b, \sin a\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \sin a\right)\right)\right)\right) \]

   10. sin-lowering-sin.f64 99.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right), \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{sin.f64}\left(a\right)\right)\right)\right)\right) \]

8. Applied egg-rr 99.6%

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

9. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

   6. +-lowering-+.f64 80.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

3. Simplified 80.2%

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

   1. cos-sum N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \left(\cos b \cdot \cos a - \color{blue}{\sin b \cdot \sin a}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\left(\cos b \cdot \cos a\right), \color{blue}{\left(\sin b \cdot \sin a\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\cos b, \cos a\right), \left(\color{blue}{\sin b} \cdot \sin a\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \cos a\right), \left(\sin \color{blue}{b} \cdot \sin a\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right), \left(\sin b \cdot \sin a\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right), \mathsf{*.f64}\left(\sin b, \color{blue}{\sin a}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \sin \color{blue}{a}\right)\right)\right) \]

   8. sin-lowering-sin.f64 99.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{sin.f64}\left(a\right)\right)\right)\right) \]

6. Applied egg-rr 99.6%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

   6. +-lowering-+.f64 80.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

3. Simplified 80.2%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{r}{\cos \left(b + a\right)}\right), \color{blue}{\sin b}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos \left(b + a\right)\right), \sin \color{blue}{b}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \sin b\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \sin b\right) \]

   8. sin-lowering-sin.f64 80.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

6. Applied egg-rr 80.2%

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

   1. cos-sum N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \left(\cos b \cdot \cos a - \sin b \cdot \sin a\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{\_.f64}\left(\left(\cos b \cdot \cos a\right), \left(\sin b \cdot \sin a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\cos b, \cos a\right), \left(\sin b \cdot \sin a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \cos a\right), \left(\sin b \cdot \sin a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right), \left(\sin b \cdot \sin a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right), \mathsf{*.f64}\left(\sin b, \sin a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \sin a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   8. sin-lowering-sin.f64 99.6%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(b\right), \mathsf{cos.f64}\left(a\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(b\right), \mathsf{sin.f64}\left(a\right)\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

8. Applied egg-rr 99.6%

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

9. Final simplification 99.6%

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

Alternative 4: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-6}:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-9}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos b}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (<= b -2e-6)
   (* r (tan b))
   (if (<= b 6.8e-9) (* b (/ r (cos a))) (/ (* r (sin b)) (cos b)))))

C

double code(double r, double a, double b) {
	double tmp;
	if (b <= -2e-6) {
		tmp = r * tan(b);
	} else if (b <= 6.8e-9) {
		tmp = b * (r / cos(a));
	} else {
		tmp = (r * sin(b)) / cos(b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-2d-6)) then
        tmp = r * tan(b)
    else if (b <= 6.8d-9) then
        tmp = b * (r / cos(a))
    else
        tmp = (r * sin(b)) / cos(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if (b <= -2e-6) {
		tmp = r * Math.tan(b);
	} else if (b <= 6.8e-9) {
		tmp = b * (r / Math.cos(a));
	} else {
		tmp = (r * Math.sin(b)) / Math.cos(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if b <= -2e-6:
		tmp = r * math.tan(b)
	elif b <= 6.8e-9:
		tmp = b * (r / math.cos(a))
	else:
		tmp = (r * math.sin(b)) / math.cos(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if (b <= -2e-6)
		tmp = Float64(r * tan(b));
	elseif (b <= 6.8e-9)
		tmp = Float64(b * Float64(r / cos(a)));
	else
		tmp = Float64(Float64(r * sin(b)) / cos(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if (b <= -2e-6)
		tmp = r * tan(b);
	elseif (b <= 6.8e-9)
		tmp = b * (r / cos(a));
	else
		tmp = (r * sin(b)) / cos(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[LessEqual[b, -2e-6], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.8e-9], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[b], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.99999999999999991e-6

   1. Initial program 53.6%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 53.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 53.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{r}{\cos \left(b + a\right)}\right), \color{blue}{\sin b}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos \left(b + a\right)\right), \sin \color{blue}{b}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \sin b\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \sin b\right) \]

      8. sin-lowering-sin.f64 53.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   6. Applied egg-rr 53.7%

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

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{r}{\cos b}\right)}, \mathsf{sin.f64}\left(b\right)\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos b\right), \mathsf{sin.f64}\left(\color{blue}{b}\right)\right) \]

      2. cos-lowering-cos.f64 54.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(b\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   9. Simplified 54.5%

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

       1. associate-*l/ N/A

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

       2. associate-/l* N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\left(\frac{\sin b}{\cos b}\right)}\right) \]

       4. quot-tan N/A

          \[\leadsto \mathsf{*.f64}\left(r, \tan b\right) \]

       5. tan-lowering-tan.f64 54.7%

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{tan.f64}\left(b\right)\right) \]

   11. Applied egg-rr 54.7%

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

   if -1.99999999999999991e-6 < b < 6.7999999999999997e-9

   1. Initial program 99.3%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 99.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 99.3%

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

   5. Taylor expanded in b around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(b \cdot r\right), \color{blue}{\cos a}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot b\right), \cos \color{blue}{a}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, b\right), \cos \color{blue}{a}\right) \]

      4. cos-lowering-cos.f64 99.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, b\right), \mathsf{cos.f64}\left(a\right)\right) \]

   7. Simplified 99.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{r}{\cos a}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{/.f64}\left(r, \color{blue}{\cos a}\right)\right) \]

      5. cos-lowering-cos.f64 99.3%

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(a\right)\right)\right) \]

   9. Applied egg-rr 99.3%

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

   if 6.7999999999999997e-9 < b

   1. Initial program 65.3%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 65.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 65.3%

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

   5. Taylor expanded in a around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos b}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{b}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos b\right) \]

      4. cos-lowering-cos.f64 65.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(b\right)\right) \]

   7. Simplified 65.8%

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

Alternative 5: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

3. Final simplification 80.2%

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

Alternative 6: 76.3% 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 80.2%

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

   6. +-lowering-+.f64 80.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

3. Simplified 80.2%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{r}{\cos \left(b + a\right)}\right), \color{blue}{\sin b}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos \left(b + a\right)\right), \sin \color{blue}{b}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \sin b\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \sin b\right) \]

   8. sin-lowering-sin.f64 80.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

6. Applied egg-rr 80.2%

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

7. Final simplification 80.2%

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

Alternative 7: 76.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \tan b\\ \mathbf{if}\;b \leq -1.1 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-9}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* r (tan b))))
   (if (<= b -1.1e-6) t_0 (if (<= b 6.8e-9) (* 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 <= -1.1e-6) {
		tmp = t_0;
	} else if (b <= 6.8e-9) {
		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 <= (-1.1d-6)) then
        tmp = t_0
    else if (b <= 6.8d-9) 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 <= -1.1e-6) {
		tmp = t_0;
	} else if (b <= 6.8e-9) {
		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 <= -1.1e-6:
		tmp = t_0
	elif b <= 6.8e-9:
		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 <= -1.1e-6)
		tmp = t_0;
	elseif (b <= 6.8e-9)
		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 <= -1.1e-6)
		tmp = t_0;
	elseif (b <= 6.8e-9)
		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, -1.1e-6], t$95$0, If[LessEqual[b, 6.8e-9], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.1000000000000001e-6 or 6.7999999999999997e-9 < b

   1. Initial program 59.5%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 59.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 59.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{r}{\cos \left(b + a\right)}\right), \color{blue}{\sin b}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos \left(b + a\right)\right), \sin \color{blue}{b}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \sin b\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \sin b\right) \]

      8. sin-lowering-sin.f64 59.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   6. Applied egg-rr 59.5%

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

   7. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{r}{\cos b}\right)}, \mathsf{sin.f64}\left(b\right)\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos b\right), \mathsf{sin.f64}\left(\color{blue}{b}\right)\right) \]

      2. cos-lowering-cos.f64 60.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(b\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

   9. Simplified 60.1%

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

       1. associate-*l/ N/A

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

       2. associate-/l* N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\left(\frac{\sin b}{\cos b}\right)}\right) \]

       4. quot-tan N/A

          \[\leadsto \mathsf{*.f64}\left(r, \tan b\right) \]

       5. tan-lowering-tan.f64 60.3%

          \[\leadsto \mathsf{*.f64}\left(r, \mathsf{tan.f64}\left(b\right)\right) \]

   11. Applied egg-rr 60.3%

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

   if -1.1000000000000001e-6 < b < 6.7999999999999997e-9

   1. Initial program 99.3%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

      6. +-lowering-+.f64 99.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

   3. Simplified 99.3%

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

   5. Taylor expanded in b around 0

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(b \cdot r\right), \color{blue}{\cos a}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(r \cdot b\right), \cos \color{blue}{a}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, b\right), \cos \color{blue}{a}\right) \]

      4. cos-lowering-cos.f64 99.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, b\right), \mathsf{cos.f64}\left(a\right)\right) \]

   7. Simplified 99.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(\frac{r}{\cos a}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{/.f64}\left(r, \color{blue}{\cos a}\right)\right) \]

      5. cos-lowering-cos.f64 99.3%

         \[\leadsto \mathsf{*.f64}\left(b, \mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(a\right)\right)\right) \]

   9. Applied egg-rr 99.3%

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

Alternative 8: 59.9% accurate, 2.0× 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 80.2%

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

   6. +-lowering-+.f64 80.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

3. Simplified 80.2%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{r}{\cos \left(b + a\right)}\right), \color{blue}{\sin b}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos \left(b + a\right)\right), \sin \color{blue}{b}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\left(b + a\right)\right)\right), \sin b\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \sin b\right) \]

   8. sin-lowering-sin.f64 80.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

6. Applied egg-rr 80.2%

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

7. Taylor expanded in a around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{r}{\cos b}\right)}, \mathsf{sin.f64}\left(b\right)\right) \]
8. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \cos b\right), \mathsf{sin.f64}\left(\color{blue}{b}\right)\right) \]

   2. cos-lowering-cos.f64 63.7%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(r, \mathsf{cos.f64}\left(b\right)\right), \mathsf{sin.f64}\left(b\right)\right) \]

9. Simplified 63.7%

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

    1. associate-*l/ N/A

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

    2. associate-/l* N/A

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

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

       \[\leadsto \mathsf{*.f64}\left(r, \color{blue}{\left(\frac{\sin b}{\cos b}\right)}\right) \]

    4. quot-tan N/A

       \[\leadsto \mathsf{*.f64}\left(r, \tan b\right) \]

    5. tan-lowering-tan.f64 63.7%

       \[\leadsto \mathsf{*.f64}\left(r, \mathsf{tan.f64}\left(b\right)\right) \]

11. Applied egg-rr 63.7%

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

Alternative 9: 34.9% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot \sin b\right), \color{blue}{\cos \left(a + b\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \sin b\right), \cos \color{blue}{\left(a + b\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \cos \left(a + \color{blue}{b}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(a + b\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\left(b + a\right)\right)\right) \]

   6. +-lowering-+.f64 80.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, \mathsf{sin.f64}\left(b\right)\right), \mathsf{cos.f64}\left(\mathsf{+.f64}\left(b, a\right)\right)\right) \]

3. Simplified 80.2%

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

5. Taylor expanded in b around 0

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(b \cdot r\right), \color{blue}{\cos a}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(r \cdot b\right), \cos \color{blue}{a}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, b\right), \cos \color{blue}{a}\right) \]

   4. cos-lowering-cos.f64 53.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(r, b\right), \mathsf{cos.f64}\left(a\right)\right) \]

7. Simplified 53.7%

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

8. Taylor expanded in a around 0

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

   1. *-lowering-*.f64 37.0%

      \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{r}\right) \]

10. Simplified 37.0%

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

11. Final simplification 37.0%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))