Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 9.5s

Alternatives: 13

Speedup: 1.0×

Specification

\[0 \leq e \land e \leq 1\]

Math

\[\begin{array}{l} \\ \frac{e \cdot \sin v}{1 + e \cdot \cos v} \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))

C

double code(double e, double v) {
	return (e * sin(v)) / (1.0 + (e * cos(v)));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = (e * sin(v)) / (1.0d0 + (e * cos(v)))
end function

Java

public static double code(double e, double v) {
	return (e * Math.sin(v)) / (1.0 + (e * Math.cos(v)));
}

Python

def code(e, v):
	return (e * math.sin(v)) / (1.0 + (e * math.cos(v)))

Julia

function code(e, v)
	return Float64(Float64(e * sin(v)) / Float64(1.0 + Float64(e * cos(v))))
end

MATLAB

function tmp = code(e, v)
	tmp = (e * sin(v)) / (1.0 + (e * cos(v)));
end

Wolfram

code[e_, v_] := N[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e \cdot \sin v}{1 + e \cdot \cos v} \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))

C

double code(double e, double v) {
	return (e * sin(v)) / (1.0 + (e * cos(v)));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = (e * sin(v)) / (1.0d0 + (e * cos(v)))
end function

Java

public static double code(double e, double v) {
	return (e * Math.sin(v)) / (1.0 + (e * Math.cos(v)));
}

Python

def code(e, v):
	return (e * math.sin(v)) / (1.0 + (e * math.cos(v)))

Julia

function code(e, v)
	return Float64(Float64(e * sin(v)) / Float64(1.0 + Float64(e * cos(v))))
end

MATLAB

function tmp = code(e, v)
	tmp = (e * sin(v)) / (1.0 + (e * cos(v)));
end

Wolfram

code[e_, v_] := N[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e \cdot \sin v}{1 + e \cdot \cos v} \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))

C

double code(double e, double v) {
	return (e * sin(v)) / (1.0 + (e * cos(v)));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = (e * sin(v)) / (1.0d0 + (e * cos(v)))
end function

Java

public static double code(double e, double v) {
	return (e * Math.sin(v)) / (1.0 + (e * Math.cos(v)));
}

Python

def code(e, v):
	return (e * math.sin(v)) / (1.0 + (e * math.cos(v)))

Julia

function code(e, v)
	return Float64(Float64(e * sin(v)) / Float64(1.0 + Float64(e * cos(v))))
end

MATLAB

function tmp = code(e, v)
	tmp = (e * sin(v)) / (1.0 + (e * cos(v)));
end

Wolfram

code[e_, v_] := N[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin v \cdot \frac{1}{\cos v + \frac{1}{e}} \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (* (sin v) (/ 1.0 (+ (cos v) (/ 1.0 e)))))

C

double code(double e, double v) {
	return sin(v) * (1.0 / (cos(v) + (1.0 / e)));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = sin(v) * (1.0d0 / (cos(v) + (1.0d0 / e)))
end function

Java

public static double code(double e, double v) {
	return Math.sin(v) * (1.0 / (Math.cos(v) + (1.0 / e)));
}

Python

def code(e, v):
	return math.sin(v) * (1.0 / (math.cos(v) + (1.0 / e)))

Julia

function code(e, v)
	return Float64(sin(v) * Float64(1.0 / Float64(cos(v) + Float64(1.0 / e))))
end

MATLAB

function tmp = code(e, v)
	tmp = sin(v) * (1.0 / (cos(v) + (1.0 / e)));
end

Wolfram

code[e_, v_] := N[(N[Sin[v], $MachinePrecision] * N[(1.0 / N[(N[Cos[v], $MachinePrecision] + N[(1.0 / e), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{e \cdot \sin v}{1 + e \cdot \cos v}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\sin v \cdot e}{\color{blue}{1} + e \cdot \cos v} \]

   2. associate-/l* N/A

      \[\leadsto \sin v \cdot \color{blue}{\frac{e}{1 + e \cdot \cos v}} \]

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

      \[\leadsto \mathsf{*.f64}\left(\sin v, \color{blue}{\left(\frac{e}{1 + e \cdot \cos v}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(v\right), \left(\frac{\color{blue}{e}}{1 + e \cdot \cos v}\right)\right) \]

   5. rgt-mult-inverse N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(v\right), \left(\frac{e}{e \cdot \frac{1}{e} + \color{blue}{e} \cdot \cos v}\right)\right) \]

   6. distribute-lft-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(v\right), \left(\frac{e}{e \cdot \color{blue}{\left(\frac{1}{e} + \cos v\right)}}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(v\right), \left(\frac{e}{e \cdot \left(\cos v + \color{blue}{\frac{1}{e}}\right)}\right)\right) \]

   8. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(v\right), \left(\frac{\frac{e}{e}}{\color{blue}{\cos v + \frac{1}{e}}}\right)\right) \]

   9. *-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(v\right), \left(\frac{\frac{e \cdot 1}{e}}{\cos \color{blue}{v} + \frac{1}{e}}\right)\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(v\right), \left(\frac{e \cdot \frac{1}{e}}{\color{blue}{\cos v} + \frac{1}{e}}\right)\right) \]

   11. rgt-mult-inverse N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(v\right), \left(\frac{1}{\color{blue}{\cos v} + \frac{1}{e}}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(v\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\cos v + \frac{1}{e}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(v\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\cos v, \color{blue}{\left(\frac{1}{e}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(v\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(v\right), \left(\frac{\color{blue}{1}}{e}\right)\right)\right)\right) \]

   15. /-lowering-/.f64 99.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(v\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(v\right), \mathsf{/.f64}\left(1, \color{blue}{e}\right)\right)\right)\right) \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\sin v \cdot \frac{1}{\cos v + \frac{1}{e}}} \]
6. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin v}{\cos v + \frac{1}{e}} \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (/ (sin v) (+ (cos v) (/ 1.0 e))))

C

double code(double e, double v) {
	return sin(v) / (cos(v) + (1.0 / e));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = sin(v) / (cos(v) + (1.0d0 / e))
end function

Java

public static double code(double e, double v) {
	return Math.sin(v) / (Math.cos(v) + (1.0 / e));
}

Python

def code(e, v):
	return math.sin(v) / (math.cos(v) + (1.0 / e))

Julia

function code(e, v)
	return Float64(sin(v) / Float64(cos(v) + Float64(1.0 / e)))
end

MATLAB

function tmp = code(e, v)
	tmp = sin(v) / (cos(v) + (1.0 / e));
end

Wolfram

code[e_, v_] := N[(N[Sin[v], $MachinePrecision] / N[(N[Cos[v], $MachinePrecision] + N[(1.0 / e), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{e \cdot \sin v}{1 + e \cdot \cos v}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\sin v \cdot e}{\color{blue}{1} + e \cdot \cos v} \]

   2. associate-/l* N/A

      \[\leadsto \sin v \cdot \color{blue}{\frac{e}{1 + e \cdot \cos v}} \]

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

      \[\leadsto \mathsf{*.f64}\left(\sin v, \color{blue}{\left(\frac{e}{1 + e \cdot \cos v}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(v\right), \left(\frac{\color{blue}{e}}{1 + e \cdot \cos v}\right)\right) \]

   5. rgt-mult-inverse N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(v\right), \left(\frac{e}{e \cdot \frac{1}{e} + \color{blue}{e} \cdot \cos v}\right)\right) \]

   6. distribute-lft-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(v\right), \left(\frac{e}{e \cdot \color{blue}{\left(\frac{1}{e} + \cos v\right)}}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(v\right), \left(\frac{e}{e \cdot \left(\cos v + \color{blue}{\frac{1}{e}}\right)}\right)\right) \]

   8. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(v\right), \left(\frac{\frac{e}{e}}{\color{blue}{\cos v + \frac{1}{e}}}\right)\right) \]

   9. *-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(v\right), \left(\frac{\frac{e \cdot 1}{e}}{\cos \color{blue}{v} + \frac{1}{e}}\right)\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(v\right), \left(\frac{e \cdot \frac{1}{e}}{\color{blue}{\cos v} + \frac{1}{e}}\right)\right) \]

   11. rgt-mult-inverse N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(v\right), \left(\frac{1}{\color{blue}{\cos v} + \frac{1}{e}}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(v\right), \mathsf{/.f64}\left(1, \color{blue}{\left(\cos v + \frac{1}{e}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(v\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\cos v, \color{blue}{\left(\frac{1}{e}\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(v\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(v\right), \left(\frac{\color{blue}{1}}{e}\right)\right)\right)\right) \]

   15. /-lowering-/.f64 99.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(v\right), \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{cos.f64}\left(v\right), \mathsf{/.f64}\left(1, \color{blue}{e}\right)\right)\right)\right) \]

5. Simplified 99.7%

   \[\leadsto \color{blue}{\sin v \cdot \frac{1}{\cos v + \frac{1}{e}}} \]

6. Taylor expanded in v around inf

   \[\leadsto \color{blue}{\frac{\sin v}{\cos v + \frac{1}{e}}} \]
7. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\sin v, \color{blue}{\left(\cos v + \frac{1}{e}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(v\right), \left(\color{blue}{\cos v} + \frac{1}{e}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(v\right), \left(\frac{1}{e} + \color{blue}{\cos v}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(v\right), \mathsf{+.f64}\left(\left(\frac{1}{e}\right), \color{blue}{\cos v}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(v\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, e\right), \cos \color{blue}{v}\right)\right) \]

   6. cos-lowering-cos.f64 99.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(v\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, e\right), \mathsf{cos.f64}\left(v\right)\right)\right) \]

8. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{\sin v}{\frac{1}{e} + \cos v}} \]

9. Final simplification 99.6%

   \[\leadsto \frac{\sin v}{\cos v + \frac{1}{e}} \]
10. Add Preprocessing

Alternative 4: 98.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ e \cdot \frac{\sin v}{e + 1} \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (* e (/ (sin v) (+ e 1.0))))

C

double code(double e, double v) {
	return e * (sin(v) / (e + 1.0));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = e * (sin(v) / (e + 1.0d0))
end function

Java

public static double code(double e, double v) {
	return e * (Math.sin(v) / (e + 1.0));
}

Python

def code(e, v):
	return e * (math.sin(v) / (e + 1.0))

Julia

function code(e, v)
	return Float64(e * Float64(sin(v) / Float64(e + 1.0)))
end

MATLAB

function tmp = code(e, v)
	tmp = e * (sin(v) / (e + 1.0));
end

Wolfram

code[e_, v_] := N[(e * N[(N[Sin[v], $MachinePrecision] / N[(e + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \color{blue}{\left(1 + e\right)}\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \left(e + \color{blue}{1}\right)\right) \]

   2. +-lowering-+.f64 98.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \mathsf{+.f64}\left(e, \color{blue}{1}\right)\right) \]

5. Simplified 98.8%

   \[\leadsto \frac{e \cdot \sin v}{\color{blue}{e + 1}} \]
6. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto e \cdot \color{blue}{\frac{\sin v}{e + 1}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin v}{e + 1} \cdot \color{blue}{e} \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin v}{e + 1}\right), \color{blue}{e}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin v, \left(e + 1\right)\right), e\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(v\right), \left(e + 1\right)\right), e\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(v\right), \left(1 + e\right)\right), e\right) \]

   7. +-lowering-+.f64 98.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(v\right), \mathsf{+.f64}\left(1, e\right)\right), e\right) \]

7. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{\sin v}{1 + e} \cdot e} \]

8. Final simplification 98.8%

   \[\leadsto e \cdot \frac{\sin v}{e + 1} \]
9. Add Preprocessing

Alternative 5: 97.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ e \cdot \sin v \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (* e (sin v)))

C

double code(double e, double v) {
	return e * sin(v);
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = e * sin(v)
end function

Java

public static double code(double e, double v) {
	return e * Math.sin(v);
}

Python

def code(e, v):
	return e * math.sin(v)

Julia

function code(e, v)
	return Float64(e * sin(v))
end

MATLAB

function tmp = code(e, v)
	tmp = e * sin(v);
end

Wolfram

code[e_, v_] := N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in e around 0

   \[\leadsto \color{blue}{e \cdot \sin v} \]
4. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(e, \color{blue}{\sin v}\right) \]

   2. sin-lowering-sin.f64 97.4%

      \[\leadsto \mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right) \]

5. Simplified 97.4%

   \[\leadsto \color{blue}{e \cdot \sin v} \]
6. Add Preprocessing

Alternative 6: 52.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.16666666666666666 + e \cdot 0.16666666666666666\\ \frac{e}{\frac{\left(e + 1\right) + v \cdot \left(v \cdot \left(e \cdot -0.5 + \left(t\_0 + \left(v \cdot v\right) \cdot \left(0.16666666666666666 \cdot \left(e \cdot -0.5 + t\_0\right) + \left(\left(e + 1\right) \cdot -0.008333333333333333 + e \cdot 0.041666666666666664\right)\right)\right)\right)\right)}{v}} \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (let* ((t_0 (+ 0.16666666666666666 (* e 0.16666666666666666))))
   (/
    e
    (/
     (+
      (+ e 1.0)
      (*
       v
       (*
        v
        (+
         (* e -0.5)
         (+
          t_0
          (*
           (* v v)
           (+
            (* 0.16666666666666666 (+ (* e -0.5) t_0))
            (+
             (* (+ e 1.0) -0.008333333333333333)
             (* e 0.041666666666666664)))))))))
     v))))

C

double code(double e, double v) {
	double t_0 = 0.16666666666666666 + (e * 0.16666666666666666);
	return e / (((e + 1.0) + (v * (v * ((e * -0.5) + (t_0 + ((v * v) * ((0.16666666666666666 * ((e * -0.5) + t_0)) + (((e + 1.0) * -0.008333333333333333) + (e * 0.041666666666666664))))))))) / v);
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    real(8) :: t_0
    t_0 = 0.16666666666666666d0 + (e * 0.16666666666666666d0)
    code = e / (((e + 1.0d0) + (v * (v * ((e * (-0.5d0)) + (t_0 + ((v * v) * ((0.16666666666666666d0 * ((e * (-0.5d0)) + t_0)) + (((e + 1.0d0) * (-0.008333333333333333d0)) + (e * 0.041666666666666664d0))))))))) / v)
end function

Java

public static double code(double e, double v) {
	double t_0 = 0.16666666666666666 + (e * 0.16666666666666666);
	return e / (((e + 1.0) + (v * (v * ((e * -0.5) + (t_0 + ((v * v) * ((0.16666666666666666 * ((e * -0.5) + t_0)) + (((e + 1.0) * -0.008333333333333333) + (e * 0.041666666666666664))))))))) / v);
}

Python

def code(e, v):
	t_0 = 0.16666666666666666 + (e * 0.16666666666666666)
	return e / (((e + 1.0) + (v * (v * ((e * -0.5) + (t_0 + ((v * v) * ((0.16666666666666666 * ((e * -0.5) + t_0)) + (((e + 1.0) * -0.008333333333333333) + (e * 0.041666666666666664))))))))) / v)

Julia

function code(e, v)
	t_0 = Float64(0.16666666666666666 + Float64(e * 0.16666666666666666))
	return Float64(e / Float64(Float64(Float64(e + 1.0) + Float64(v * Float64(v * Float64(Float64(e * -0.5) + Float64(t_0 + Float64(Float64(v * v) * Float64(Float64(0.16666666666666666 * Float64(Float64(e * -0.5) + t_0)) + Float64(Float64(Float64(e + 1.0) * -0.008333333333333333) + Float64(e * 0.041666666666666664))))))))) / v))
end

MATLAB

function tmp = code(e, v)
	t_0 = 0.16666666666666666 + (e * 0.16666666666666666);
	tmp = e / (((e + 1.0) + (v * (v * ((e * -0.5) + (t_0 + ((v * v) * ((0.16666666666666666 * ((e * -0.5) + t_0)) + (((e + 1.0) * -0.008333333333333333) + (e * 0.041666666666666664))))))))) / v);
end

Wolfram

code[e_, v_] := Block[{t$95$0 = N[(0.16666666666666666 + N[(e * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, N[(e / N[(N[(N[(e + 1.0), $MachinePrecision] + N[(v * N[(v * N[(N[(e * -0.5), $MachinePrecision] + N[(t$95$0 + N[(N[(v * v), $MachinePrecision] * N[(N[(0.16666666666666666 * N[(N[(e * -0.5), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(e + 1.0), $MachinePrecision] * -0.008333333333333333), $MachinePrecision] + N[(e * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + e \cdot \cos v}{e \cdot \sin v}}} \]

   2. inv-pow N/A

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

   3. sqr-pow N/A

      \[\leadsto {\left(\frac{1 + e \cdot \cos v}{e \cdot \sin v}\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\frac{1 + e \cdot \cos v}{e \cdot \sin v}\right)}^{\left(\frac{-1}{2}\right)}} \]

   4. pow2 N/A

      \[\leadsto {\left({\left(\frac{1 + e \cdot \cos v}{e \cdot \sin v}\right)}^{\left(\frac{-1}{2}\right)}\right)}^{\color{blue}{2}} \]

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

      \[\leadsto \mathsf{pow.f64}\left(\left({\left(\frac{1 + e \cdot \cos v}{e \cdot \sin v}\right)}^{\left(\frac{-1}{2}\right)}\right), \color{blue}{2}\right) \]

4. Applied egg-rr 61.0%

   \[\leadsto \color{blue}{{\left({\left(\frac{e}{\frac{1 + e \cdot \cos v}{\sin v}}\right)}^{0.5}\right)}^{2}} \]
5. Step-by-step derivation

   1. pow-pow N/A

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

   2. metadata-eval N/A

      \[\leadsto {\left(\frac{e}{\frac{1 + e \cdot \cos v}{\sin v}}\right)}^{1} \]

   3. unpow1 N/A

      \[\leadsto \frac{e}{\color{blue}{\frac{1 + e \cdot \cos v}{\sin v}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(e, \color{blue}{\left(\frac{1 + e \cdot \cos v}{\sin v}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\left(1 + e \cdot \cos v\right), \color{blue}{\sin v}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(e \cdot \cos v\right)\right), \sin \color{blue}{v}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\cos v \cdot e\right)\right), \sin v\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\cos v, e\right)\right), \sin v\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(v\right), e\right)\right), \sin v\right)\right) \]

   10. sin-lowering-sin.f64 99.7%

       \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(v\right), e\right)\right), \mathsf{sin.f64}\left(v\right)\right)\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{e}{\frac{1 + \cos v \cdot e}{\sin v}}} \]

7. Taylor expanded in v around 0

   \[\leadsto \mathsf{/.f64}\left(e, \color{blue}{\left(\frac{1 + \left(e + {v}^{2} \cdot \left(\left(\frac{-1}{2} \cdot e + {v}^{2} \cdot \left(\frac{1}{24} \cdot e - \left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right) + \frac{1}{120} \cdot \left(1 + e\right)\right)\right)\right) - \frac{-1}{6} \cdot \left(1 + e\right)\right)\right)}{v}\right)}\right) \]

9. Simplified 54.7%

   \[\leadsto \frac{e}{\color{blue}{\frac{\left(e + 1\right) + v \cdot \left(v \cdot \left(e \cdot -0.5 + \left(\left(v \cdot v\right) \cdot \left(\left(e \cdot -0.5 + \left(0.16666666666666666 + e \cdot 0.16666666666666666\right)\right) \cdot 0.16666666666666666 + \left(\left(e + 1\right) \cdot -0.008333333333333333 + e \cdot 0.041666666666666664\right)\right) + \left(0.16666666666666666 + e \cdot 0.16666666666666666\right)\right)\right)\right)}{v}}} \]

10. Final simplification 54.7%

    \[\leadsto \frac{e}{\frac{\left(e + 1\right) + v \cdot \left(v \cdot \left(e \cdot -0.5 + \left(\left(0.16666666666666666 + e \cdot 0.16666666666666666\right) + \left(v \cdot v\right) \cdot \left(0.16666666666666666 \cdot \left(e \cdot -0.5 + \left(0.16666666666666666 + e \cdot 0.16666666666666666\right)\right) + \left(\left(e + 1\right) \cdot -0.008333333333333333 + e \cdot 0.041666666666666664\right)\right)\right)\right)\right)}{v}} \]
11. Add Preprocessing

Alternative 7: 52.4% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ e \cdot \frac{v}{e + \left(1 + v \cdot \left(v \cdot \left(e \cdot -0.5 + \left(e + 1\right) \cdot 0.16666666666666666\right)\right)\right)} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (*
  e
  (/
   v
   (+
    e
    (+ 1.0 (* v (* v (+ (* e -0.5) (* (+ e 1.0) 0.16666666666666666)))))))))

C

double code(double e, double v) {
	return e * (v / (e + (1.0 + (v * (v * ((e * -0.5) + ((e + 1.0) * 0.16666666666666666)))))));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = e * (v / (e + (1.0d0 + (v * (v * ((e * (-0.5d0)) + ((e + 1.0d0) * 0.16666666666666666d0)))))))
end function

Java

public static double code(double e, double v) {
	return e * (v / (e + (1.0 + (v * (v * ((e * -0.5) + ((e + 1.0) * 0.16666666666666666)))))));
}

Python

def code(e, v):
	return e * (v / (e + (1.0 + (v * (v * ((e * -0.5) + ((e + 1.0) * 0.16666666666666666)))))))

Julia

function code(e, v)
	return Float64(e * Float64(v / Float64(e + Float64(1.0 + Float64(v * Float64(v * Float64(Float64(e * -0.5) + Float64(Float64(e + 1.0) * 0.16666666666666666))))))))
end

MATLAB

function tmp = code(e, v)
	tmp = e * (v / (e + (1.0 + (v * (v * ((e * -0.5) + ((e + 1.0) * 0.16666666666666666)))))));
end

Wolfram

code[e_, v_] := N[(e * N[(v / N[(e + N[(1.0 + N[(v * N[(v * N[(N[(e * -0.5), $MachinePrecision] + N[(N[(e + 1.0), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + e \cdot \cos v}{e \cdot \sin v}}} \]

   2. inv-pow N/A

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

   3. sqr-pow N/A

      \[\leadsto {\left(\frac{1 + e \cdot \cos v}{e \cdot \sin v}\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\frac{1 + e \cdot \cos v}{e \cdot \sin v}\right)}^{\left(\frac{-1}{2}\right)}} \]

   4. pow2 N/A

      \[\leadsto {\left({\left(\frac{1 + e \cdot \cos v}{e \cdot \sin v}\right)}^{\left(\frac{-1}{2}\right)}\right)}^{\color{blue}{2}} \]

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

      \[\leadsto \mathsf{pow.f64}\left(\left({\left(\frac{1 + e \cdot \cos v}{e \cdot \sin v}\right)}^{\left(\frac{-1}{2}\right)}\right), \color{blue}{2}\right) \]

4. Applied egg-rr 61.0%

   \[\leadsto \color{blue}{{\left({\left(\frac{e}{\frac{1 + e \cdot \cos v}{\sin v}}\right)}^{0.5}\right)}^{2}} \]
5. Step-by-step derivation

   1. pow-pow N/A

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

   2. metadata-eval N/A

      \[\leadsto {\left(\frac{e}{\frac{1 + e \cdot \cos v}{\sin v}}\right)}^{1} \]

   3. unpow1 N/A

      \[\leadsto \frac{e}{\color{blue}{\frac{1 + e \cdot \cos v}{\sin v}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(e, \color{blue}{\left(\frac{1 + e \cdot \cos v}{\sin v}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\left(1 + e \cdot \cos v\right), \color{blue}{\sin v}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(e \cdot \cos v\right)\right), \sin \color{blue}{v}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\cos v \cdot e\right)\right), \sin v\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\cos v, e\right)\right), \sin v\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(v\right), e\right)\right), \sin v\right)\right) \]

   10. sin-lowering-sin.f64 99.7%

       \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(v\right), e\right)\right), \mathsf{sin.f64}\left(v\right)\right)\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{e}{\frac{1 + \cos v \cdot e}{\sin v}}} \]

7. Taylor expanded in v around 0

   \[\leadsto \mathsf{/.f64}\left(e, \color{blue}{\left(\frac{1 + \left(e + {v}^{2} \cdot \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right)\right)}{v}\right)}\right) \]
8. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\left(1 + \left(e + {v}^{2} \cdot \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right)\right)\right), \color{blue}{v}\right)\right) \]

9. Simplified 54.6%

   \[\leadsto \frac{e}{\color{blue}{\frac{\left(e + 1\right) + v \cdot \left(v \cdot \left(e \cdot -0.5 + \left(0.16666666666666666 + e \cdot 0.16666666666666666\right)\right)\right)}{v}}} \]
10. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\left(e + 1\right) + v \cdot \left(v \cdot \left(e \cdot \frac{-1}{2} + \left(\frac{1}{6} + e \cdot \frac{1}{6}\right)\right)\right)}{v}}{e}}} \]

    2. associate-/r/ N/A

       \[\leadsto \frac{1}{\frac{\left(e + 1\right) + v \cdot \left(v \cdot \left(e \cdot \frac{-1}{2} + \left(\frac{1}{6} + e \cdot \frac{1}{6}\right)\right)\right)}{v}} \cdot \color{blue}{e} \]

    3. clear-num N/A

       \[\leadsto \frac{v}{\left(e + 1\right) + v \cdot \left(v \cdot \left(e \cdot \frac{-1}{2} + \left(\frac{1}{6} + e \cdot \frac{1}{6}\right)\right)\right)} \cdot e \]

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

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{v}{\left(e + 1\right) + v \cdot \left(v \cdot \left(e \cdot \frac{-1}{2} + \left(\frac{1}{6} + e \cdot \frac{1}{6}\right)\right)\right)}\right), \color{blue}{e}\right) \]

11. Applied egg-rr 54.7%

    \[\leadsto \color{blue}{\frac{v}{e + \left(1 + v \cdot \left(v \cdot \left(e \cdot -0.5 + \left(1 + e\right) \cdot 0.16666666666666666\right)\right)\right)} \cdot e} \]

12. Final simplification 54.7%

    \[\leadsto e \cdot \frac{v}{e + \left(1 + v \cdot \left(v \cdot \left(e \cdot -0.5 + \left(e + 1\right) \cdot 0.16666666666666666\right)\right)\right)} \]
13. Add Preprocessing

Alternative 8: 52.4% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ v \cdot \frac{e}{e + \left(1 + v \cdot \left(v \cdot \left(e \cdot -0.5 + \left(e + 1\right) \cdot 0.16666666666666666\right)\right)\right)} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (*
  v
  (/
   e
   (+
    e
    (+ 1.0 (* v (* v (+ (* e -0.5) (* (+ e 1.0) 0.16666666666666666)))))))))

C

double code(double e, double v) {
	return v * (e / (e + (1.0 + (v * (v * ((e * -0.5) + ((e + 1.0) * 0.16666666666666666)))))));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = v * (e / (e + (1.0d0 + (v * (v * ((e * (-0.5d0)) + ((e + 1.0d0) * 0.16666666666666666d0)))))))
end function

Java

public static double code(double e, double v) {
	return v * (e / (e + (1.0 + (v * (v * ((e * -0.5) + ((e + 1.0) * 0.16666666666666666)))))));
}

Python

def code(e, v):
	return v * (e / (e + (1.0 + (v * (v * ((e * -0.5) + ((e + 1.0) * 0.16666666666666666)))))))

Julia

function code(e, v)
	return Float64(v * Float64(e / Float64(e + Float64(1.0 + Float64(v * Float64(v * Float64(Float64(e * -0.5) + Float64(Float64(e + 1.0) * 0.16666666666666666))))))))
end

MATLAB

function tmp = code(e, v)
	tmp = v * (e / (e + (1.0 + (v * (v * ((e * -0.5) + ((e + 1.0) * 0.16666666666666666)))))));
end

Wolfram

code[e_, v_] := N[(v * N[(e / N[(e + N[(1.0 + N[(v * N[(v * N[(N[(e * -0.5), $MachinePrecision] + N[(N[(e + 1.0), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + e \cdot \cos v}{e \cdot \sin v}}} \]

   2. inv-pow N/A

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

   3. sqr-pow N/A

      \[\leadsto {\left(\frac{1 + e \cdot \cos v}{e \cdot \sin v}\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\frac{1 + e \cdot \cos v}{e \cdot \sin v}\right)}^{\left(\frac{-1}{2}\right)}} \]

   4. pow2 N/A

      \[\leadsto {\left({\left(\frac{1 + e \cdot \cos v}{e \cdot \sin v}\right)}^{\left(\frac{-1}{2}\right)}\right)}^{\color{blue}{2}} \]

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

      \[\leadsto \mathsf{pow.f64}\left(\left({\left(\frac{1 + e \cdot \cos v}{e \cdot \sin v}\right)}^{\left(\frac{-1}{2}\right)}\right), \color{blue}{2}\right) \]

4. Applied egg-rr 61.0%

   \[\leadsto \color{blue}{{\left({\left(\frac{e}{\frac{1 + e \cdot \cos v}{\sin v}}\right)}^{0.5}\right)}^{2}} \]
5. Step-by-step derivation

   1. pow-pow N/A

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

   2. metadata-eval N/A

      \[\leadsto {\left(\frac{e}{\frac{1 + e \cdot \cos v}{\sin v}}\right)}^{1} \]

   3. unpow1 N/A

      \[\leadsto \frac{e}{\color{blue}{\frac{1 + e \cdot \cos v}{\sin v}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(e, \color{blue}{\left(\frac{1 + e \cdot \cos v}{\sin v}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\left(1 + e \cdot \cos v\right), \color{blue}{\sin v}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(e \cdot \cos v\right)\right), \sin \color{blue}{v}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\cos v \cdot e\right)\right), \sin v\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\cos v, e\right)\right), \sin v\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(v\right), e\right)\right), \sin v\right)\right) \]

   10. sin-lowering-sin.f64 99.7%

       \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(v\right), e\right)\right), \mathsf{sin.f64}\left(v\right)\right)\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{e}{\frac{1 + \cos v \cdot e}{\sin v}}} \]

7. Taylor expanded in v around 0

   \[\leadsto \mathsf{/.f64}\left(e, \color{blue}{\left(\frac{1 + \left(e + {v}^{2} \cdot \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right)\right)}{v}\right)}\right) \]
8. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\left(1 + \left(e + {v}^{2} \cdot \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right)\right)\right), \color{blue}{v}\right)\right) \]

9. Simplified 54.6%

   \[\leadsto \frac{e}{\color{blue}{\frac{\left(e + 1\right) + v \cdot \left(v \cdot \left(e \cdot -0.5 + \left(0.16666666666666666 + e \cdot 0.16666666666666666\right)\right)\right)}{v}}} \]
10. Step-by-step derivation

    1. associate-/r/ N/A

       \[\leadsto \frac{e}{\left(e + 1\right) + v \cdot \left(v \cdot \left(e \cdot \frac{-1}{2} + \left(\frac{1}{6} + e \cdot \frac{1}{6}\right)\right)\right)} \cdot \color{blue}{v} \]

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

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{e}{\left(e + 1\right) + v \cdot \left(v \cdot \left(e \cdot \frac{-1}{2} + \left(\frac{1}{6} + e \cdot \frac{1}{6}\right)\right)\right)}\right), \color{blue}{v}\right) \]

11. Applied egg-rr 54.6%

    \[\leadsto \color{blue}{\frac{e}{e + \left(1 + v \cdot \left(v \cdot \left(e \cdot -0.5 + \left(1 + e\right) \cdot 0.16666666666666666\right)\right)\right)} \cdot v} \]

12. Final simplification 54.6%

    \[\leadsto v \cdot \frac{e}{e + \left(1 + v \cdot \left(v \cdot \left(e \cdot -0.5 + \left(e + 1\right) \cdot 0.16666666666666666\right)\right)\right)} \]
13. Add Preprocessing

Alternative 9: 52.0% accurate, 13.9× speedup

Math

\[\begin{array}{l} \\ \frac{e}{\frac{\left(e + 1\right) + -0.3333333333333333 \cdot \left(e \cdot \left(v \cdot v\right)\right)}{v}} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (/ e (/ (+ (+ e 1.0) (* -0.3333333333333333 (* e (* v v)))) v)))

C

double code(double e, double v) {
	return e / (((e + 1.0) + (-0.3333333333333333 * (e * (v * v)))) / v);
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = e / (((e + 1.0d0) + ((-0.3333333333333333d0) * (e * (v * v)))) / v)
end function

Java

public static double code(double e, double v) {
	return e / (((e + 1.0) + (-0.3333333333333333 * (e * (v * v)))) / v);
}

Python

def code(e, v):
	return e / (((e + 1.0) + (-0.3333333333333333 * (e * (v * v)))) / v)

Julia

function code(e, v)
	return Float64(e / Float64(Float64(Float64(e + 1.0) + Float64(-0.3333333333333333 * Float64(e * Float64(v * v)))) / v))
end

MATLAB

function tmp = code(e, v)
	tmp = e / (((e + 1.0) + (-0.3333333333333333 * (e * (v * v)))) / v);
end

Wolfram

code[e_, v_] := N[(e / N[(N[(N[(e + 1.0), $MachinePrecision] + N[(-0.3333333333333333 * N[(e * N[(v * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + e \cdot \cos v}{e \cdot \sin v}}} \]

   2. inv-pow N/A

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

   3. sqr-pow N/A

      \[\leadsto {\left(\frac{1 + e \cdot \cos v}{e \cdot \sin v}\right)}^{\left(\frac{-1}{2}\right)} \cdot \color{blue}{{\left(\frac{1 + e \cdot \cos v}{e \cdot \sin v}\right)}^{\left(\frac{-1}{2}\right)}} \]

   4. pow2 N/A

      \[\leadsto {\left({\left(\frac{1 + e \cdot \cos v}{e \cdot \sin v}\right)}^{\left(\frac{-1}{2}\right)}\right)}^{\color{blue}{2}} \]

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

      \[\leadsto \mathsf{pow.f64}\left(\left({\left(\frac{1 + e \cdot \cos v}{e \cdot \sin v}\right)}^{\left(\frac{-1}{2}\right)}\right), \color{blue}{2}\right) \]

4. Applied egg-rr 61.0%

   \[\leadsto \color{blue}{{\left({\left(\frac{e}{\frac{1 + e \cdot \cos v}{\sin v}}\right)}^{0.5}\right)}^{2}} \]
5. Step-by-step derivation

   1. pow-pow N/A

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

   2. metadata-eval N/A

      \[\leadsto {\left(\frac{e}{\frac{1 + e \cdot \cos v}{\sin v}}\right)}^{1} \]

   3. unpow1 N/A

      \[\leadsto \frac{e}{\color{blue}{\frac{1 + e \cdot \cos v}{\sin v}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(e, \color{blue}{\left(\frac{1 + e \cdot \cos v}{\sin v}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\left(1 + e \cdot \cos v\right), \color{blue}{\sin v}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(e \cdot \cos v\right)\right), \sin \color{blue}{v}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\cos v \cdot e\right)\right), \sin v\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\cos v, e\right)\right), \sin v\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(v\right), e\right)\right), \sin v\right)\right) \]

   10. sin-lowering-sin.f64 99.7%

       \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(v\right), e\right)\right), \mathsf{sin.f64}\left(v\right)\right)\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{e}{\frac{1 + \cos v \cdot e}{\sin v}}} \]

7. Taylor expanded in v around 0

   \[\leadsto \mathsf{/.f64}\left(e, \color{blue}{\left(\frac{1 + \left(e + {v}^{2} \cdot \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right)\right)}{v}\right)}\right) \]
8. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\left(1 + \left(e + {v}^{2} \cdot \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right)\right)\right), \color{blue}{v}\right)\right) \]

9. Simplified 54.6%

   \[\leadsto \frac{e}{\color{blue}{\frac{\left(e + 1\right) + v \cdot \left(v \cdot \left(e \cdot -0.5 + \left(0.16666666666666666 + e \cdot 0.16666666666666666\right)\right)\right)}{v}}} \]

10. Taylor expanded in e around inf

    \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(e, 1\right), \color{blue}{\left(\frac{-1}{3} \cdot \left(e \cdot {v}^{2}\right)\right)}\right), v\right)\right) \]
11. Step-by-step derivation

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

       \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(e, 1\right), \mathsf{*.f64}\left(\frac{-1}{3}, \left(e \cdot {v}^{2}\right)\right)\right), v\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(e, 1\right), \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(e, \left({v}^{2}\right)\right)\right)\right), v\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(e, 1\right), \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(e, \left(v \cdot v\right)\right)\right)\right), v\right)\right) \]

    4. *-lowering-*.f64 54.3%

       \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(e, 1\right), \mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(e, \mathsf{*.f64}\left(v, v\right)\right)\right)\right), v\right)\right) \]

12. Simplified 54.3%

    \[\leadsto \frac{e}{\frac{\left(e + 1\right) + \color{blue}{-0.3333333333333333 \cdot \left(e \cdot \left(v \cdot v\right)\right)}}{v}} \]
13. Add Preprocessing

Alternative 10: 51.2% accurate, 29.9× speedup

Math

\[\begin{array}{l} \\ e \cdot \frac{v}{e + 1} \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (* e (/ v (+ e 1.0))))

C

double code(double e, double v) {
	return e * (v / (e + 1.0));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = e * (v / (e + 1.0d0))
end function

Java

public static double code(double e, double v) {
	return e * (v / (e + 1.0));
}

Python

def code(e, v):
	return e * (v / (e + 1.0))

Julia

function code(e, v)
	return Float64(e * Float64(v / Float64(e + 1.0)))
end

MATLAB

function tmp = code(e, v)
	tmp = e * (v / (e + 1.0));
end

Wolfram

code[e_, v_] := N[(e * N[(v / N[(e + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto e \cdot \color{blue}{\frac{v}{1 + e}} \]

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

      \[\leadsto \mathsf{*.f64}\left(e, \color{blue}{\left(\frac{v}{1 + e}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(e, \mathsf{/.f64}\left(v, \color{blue}{\left(1 + e\right)}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(e, \mathsf{/.f64}\left(v, \left(e + \color{blue}{1}\right)\right)\right) \]

   5. +-lowering-+.f64 53.6%

      \[\leadsto \mathsf{*.f64}\left(e, \mathsf{/.f64}\left(v, \mathsf{+.f64}\left(e, \color{blue}{1}\right)\right)\right) \]

5. Simplified 53.6%

   \[\leadsto \color{blue}{e \cdot \frac{v}{e + 1}} \]
6. Add Preprocessing

Alternative 11: 50.8% accurate, 29.9× speedup

Math

\[\begin{array}{l} \\ e \cdot \left(v - e \cdot v\right) \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (* e (- v (* e v))))

C

double code(double e, double v) {
	return e * (v - (e * v));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = e * (v - (e * v))
end function

Java

public static double code(double e, double v) {
	return e * (v - (e * v));
}

Python

def code(e, v):
	return e * (v - (e * v))

Julia

function code(e, v)
	return Float64(e * Float64(v - Float64(e * v)))
end

MATLAB

function tmp = code(e, v)
	tmp = e * (v - (e * v));
end

Wolfram

code[e_, v_] := N[(e * N[(v - N[(e * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto e \cdot \color{blue}{\frac{v}{1 + e}} \]

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

      \[\leadsto \mathsf{*.f64}\left(e, \color{blue}{\left(\frac{v}{1 + e}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(e, \mathsf{/.f64}\left(v, \color{blue}{\left(1 + e\right)}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(e, \mathsf{/.f64}\left(v, \left(e + \color{blue}{1}\right)\right)\right) \]

   5. +-lowering-+.f64 53.6%

      \[\leadsto \mathsf{*.f64}\left(e, \mathsf{/.f64}\left(v, \mathsf{+.f64}\left(e, \color{blue}{1}\right)\right)\right) \]

5. Simplified 53.6%

   \[\leadsto \color{blue}{e \cdot \frac{v}{e + 1}} \]

6. Taylor expanded in e around 0

   \[\leadsto \color{blue}{e \cdot \left(v + -1 \cdot \left(e \cdot v\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto e \cdot \left(-1 \cdot \left(e \cdot v\right) + \color{blue}{v}\right) \]

   2. remove-double-neg N/A

      \[\leadsto e \cdot \left(-1 \cdot \left(e \cdot v\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(v\right)\right)\right)\right)\right) \]

   3. sub-neg N/A

      \[\leadsto e \cdot \left(-1 \cdot \left(e \cdot v\right) - \color{blue}{\left(\mathsf{neg}\left(v\right)\right)}\right) \]

   4. mul-1-neg N/A

      \[\leadsto e \cdot \left(-1 \cdot \left(e \cdot v\right) - -1 \cdot \color{blue}{v}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(e, \color{blue}{\left(-1 \cdot \left(e \cdot v\right) - -1 \cdot v\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(e, \left(-1 \cdot \left(e \cdot v\right) + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot v}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(e, \left(-1 \cdot \left(e \cdot v\right) + 1 \cdot v\right)\right) \]

   8. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(e, \left(-1 \cdot \left(e \cdot v\right) + v\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(e, \left(v + \color{blue}{-1 \cdot \left(e \cdot v\right)}\right)\right) \]

   10. mul-1-neg N/A

       \[\leadsto \mathsf{*.f64}\left(e, \left(v + \left(\mathsf{neg}\left(e \cdot v\right)\right)\right)\right) \]

   11. unsub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(e, \left(v - \color{blue}{e \cdot v}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(e, \mathsf{\_.f64}\left(v, \color{blue}{\left(e \cdot v\right)}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(e, \mathsf{\_.f64}\left(v, \left(v \cdot \color{blue}{e}\right)\right)\right) \]

   14. *-lowering-*.f64 52.9%

       \[\leadsto \mathsf{*.f64}\left(e, \mathsf{\_.f64}\left(v, \mathsf{*.f64}\left(v, \color{blue}{e}\right)\right)\right) \]

8. Simplified 52.9%

   \[\leadsto \color{blue}{e \cdot \left(v - v \cdot e\right)} \]

9. Final simplification 52.9%

   \[\leadsto e \cdot \left(v - e \cdot v\right) \]
10. Add Preprocessing

Alternative 12: 50.3% accurate, 69.7× speedup

Math

\[\begin{array}{l} \\ e \cdot v \end{array} \]

FPCore

(FPCore (e v) :precision binary64 (* e v))

C

double code(double e, double v) {
	return e * v;
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = e * v
end function

Java

public static double code(double e, double v) {
	return e * v;
}

Python

def code(e, v):
	return e * v

Julia

function code(e, v)
	return Float64(e * v)
end

MATLAB

function tmp = code(e, v)
	tmp = e * v;
end

Wolfram

code[e_, v_] := N[(e * v), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto e \cdot \color{blue}{\frac{v}{1 + e}} \]

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

      \[\leadsto \mathsf{*.f64}\left(e, \color{blue}{\left(\frac{v}{1 + e}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(e, \mathsf{/.f64}\left(v, \color{blue}{\left(1 + e\right)}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(e, \mathsf{/.f64}\left(v, \left(e + \color{blue}{1}\right)\right)\right) \]

   5. +-lowering-+.f64 53.6%

      \[\leadsto \mathsf{*.f64}\left(e, \mathsf{/.f64}\left(v, \mathsf{+.f64}\left(e, \color{blue}{1}\right)\right)\right) \]

5. Simplified 53.6%

   \[\leadsto \color{blue}{e \cdot \frac{v}{e + 1}} \]

6. Taylor expanded in e around 0

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

   1. *-commutative N/A

      \[\leadsto v \cdot \color{blue}{e} \]

   2. *-lowering-*.f64 52.1%

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

8. Simplified 52.1%

   \[\leadsto \color{blue}{v \cdot e} \]

9. Final simplification 52.1%

   \[\leadsto e \cdot v \]
10. Add Preprocessing

Alternative 13: 4.5% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ v \end{array} \]

FPCore

(FPCore (e v) :precision binary64 v)

C

double code(double e, double v) {
	return v;
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = v
end function

Java

public static double code(double e, double v) {
	return v;
}

Python

def code(e, v):
	return v

Julia

function code(e, v)
	return v
end

MATLAB

function tmp = code(e, v)
	tmp = v;
end

Wolfram

code[e_, v_] := v

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]
4. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto e \cdot \color{blue}{\frac{v}{1 + e}} \]

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

      \[\leadsto \mathsf{*.f64}\left(e, \color{blue}{\left(\frac{v}{1 + e}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(e, \mathsf{/.f64}\left(v, \color{blue}{\left(1 + e\right)}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(e, \mathsf{/.f64}\left(v, \left(e + \color{blue}{1}\right)\right)\right) \]

   5. +-lowering-+.f64 53.6%

      \[\leadsto \mathsf{*.f64}\left(e, \mathsf{/.f64}\left(v, \mathsf{+.f64}\left(e, \color{blue}{1}\right)\right)\right) \]

5. Simplified 53.6%

   \[\leadsto \color{blue}{e \cdot \frac{v}{e + 1}} \]

6. Taylor expanded in e around inf

   \[\leadsto \color{blue}{v} \]
7. Step-by-step derivation

8. Simplified 4.6%

   \[\leadsto \color{blue}{v} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))