Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 9.3s

Alternatives: 12

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

   8. sin-lowering-sin.f64 99.6%

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

4. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{e}{\frac{1 + e \cdot \cos v}{\sin v}}} \]
5. 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. rgt-mult-inverse N/A

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

   2. distribute-lft-in N/A

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

   3. +-commutative N/A

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

   4. times-frac N/A

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

   5. *-rgt-identity N/A

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

   6. associate-*r/ N/A

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

   7. rgt-mult-inverse N/A

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

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

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

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

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

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

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

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

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

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

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

   13. /-lowering-/.f64 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 4: 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 98.5%

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

5. Simplified 98.5%

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

Alternative 5: 52.2% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \frac{v}{\left(1 + \frac{1}{e}\right) - \left(v \cdot v\right) \cdot \left(\frac{-0.16666666666666666}{e} + 0.3333333333333333\right)} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (/
  v
  (-
   (+ 1.0 (/ 1.0 e))
   (* (* v v) (+ (/ -0.16666666666666666 e) 0.3333333333333333)))))

C

double code(double e, double v) {
	return v / ((1.0 + (1.0 / e)) - ((v * v) * ((-0.16666666666666666 / e) + 0.3333333333333333)));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = v / ((1.0d0 + (1.0d0 / e)) - ((v * v) * (((-0.16666666666666666d0) / e) + 0.3333333333333333d0)))
end function

Java

public static double code(double e, double v) {
	return v / ((1.0 + (1.0 / e)) - ((v * v) * ((-0.16666666666666666 / e) + 0.3333333333333333)));
}

Python

def code(e, v):
	return v / ((1.0 + (1.0 / e)) - ((v * v) * ((-0.16666666666666666 / e) + 0.3333333333333333)))

Julia

function code(e, v)
	return Float64(v / Float64(Float64(1.0 + Float64(1.0 / e)) - Float64(Float64(v * v) * Float64(Float64(-0.16666666666666666 / e) + 0.3333333333333333))))
end

MATLAB

function tmp = code(e, v)
	tmp = v / ((1.0 + (1.0 / e)) - ((v * v) * ((-0.16666666666666666 / e) + 0.3333333333333333)));
end

Wolfram

code[e_, v_] := N[(v / N[(N[(1.0 + N[(1.0 / e), $MachinePrecision]), $MachinePrecision] - N[(N[(v * v), $MachinePrecision] * N[(N[(-0.16666666666666666 / e), $MachinePrecision] + 0.3333333333333333), $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. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

   8. sin-lowering-sin.f64 99.2%

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

4. Applied egg-rr 99.2%

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

5. Taylor expanded in v around 0

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

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

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

7. Simplified 51.2%

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

9. Applied egg-rr 51.7%

   \[\leadsto \color{blue}{\frac{v}{\left(1 + \frac{1}{e}\right) - \left(v \cdot v\right) \cdot \left(\frac{-0.16666666666666666}{e} + 0.3333333333333333\right)}} \]
10. Add Preprocessing

Alternative 6: 50.8% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(1.0 / N[(N[(1.0 + N[(N[(1.0 / e), $MachinePrecision] - N[(N[(v * v), $MachinePrecision] * N[(-0.16666666666666666 + 0.5), $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. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

   8. sin-lowering-sin.f64 99.2%

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

4. Applied egg-rr 99.2%

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

5. Taylor expanded in v around 0

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

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

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

7. Simplified 51.2%

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

8. Taylor expanded in e around inf

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

10. Simplified 51.0%

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

11. Final simplification 51.0%

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

Alternative 7: 51.3% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(e * v), $MachinePrecision] / N[(1.0 + N[(N[(v * v), $MachinePrecision] * 0.16666666666666666), $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. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

   8. sin-lowering-sin.f64 99.2%

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

4. Applied egg-rr 99.2%

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

5. Taylor expanded in v around 0

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

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

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

7. Simplified 51.2%

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

8. Taylor expanded in e around 0

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 - \frac{-1}{6} \cdot {v}^{2}}} \]
9. Step-by-step derivation

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

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

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

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

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

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

   4. metadata-eval N/A

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

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

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

   6. *-commutative N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 50.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, v\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \frac{1}{6}\right)\right)\right) \]

10. Simplified 50.9%

    \[\leadsto \color{blue}{\frac{e \cdot v}{1 + \left(v \cdot v\right) \cdot 0.16666666666666666}} \]
11. Add Preprocessing

Alternative 8: 51.3% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e * N[(v / N[(1.0 + N[(v * N[(v * 0.16666666666666666), $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. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

   8. sin-lowering-sin.f64 99.2%

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

4. Applied egg-rr 99.2%

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

5. Taylor expanded in v around 0

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

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

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

7. Simplified 51.2%

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

8. Taylor expanded in e around 0

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

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

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

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

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

   3. metadata-eval N/A

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

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

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

   5. *-commutative N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. *-lowering-*.f64 50.4%

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

10. Simplified 50.4%

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

    1. associate-/r/ N/A

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

    2. *-commutative N/A

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

    3. associate-*r* N/A

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

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

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

    5. associate-*l/ N/A

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

    6. *-lft-identity N/A

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

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

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

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

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

    9. associate-*l* N/A

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

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

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

    11. *-lowering-*.f64 50.9%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \frac{1}{6}\right)\right)\right)\right), e\right) \]

12. Applied egg-rr 50.9%

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

13. Final simplification 50.9%

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

Alternative 9: 51.3% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(v * N[(e / N[(1.0 + N[(v * N[(v * 0.16666666666666666), $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. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

   8. sin-lowering-sin.f64 99.2%

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

4. Applied egg-rr 99.2%

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

5. Taylor expanded in v around 0

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

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

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

7. Simplified 51.2%

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

8. Taylor expanded in e around 0

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

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

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

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

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

   3. metadata-eval N/A

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

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

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

   5. *-commutative N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. *-lowering-*.f64 50.4%

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

10. Simplified 50.4%

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

    1. associate-/r* N/A

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

    2. associate-/r/ N/A

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

    3. clear-num N/A

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

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

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

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

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

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

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

    7. associate-*l* N/A

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

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

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

    9. *-lowering-*.f64 50.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(e, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(v, \mathsf{*.f64}\left(v, \frac{1}{6}\right)\right)\right)\right), v\right) \]

12. Applied egg-rr 50.9%

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

13. Final simplification 50.9%

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

Alternative 10: 51.1% 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. /-lowering-/.f64 N/A

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

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

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

   3. +-commutative N/A

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

   4. +-lowering-+.f64 50.7%

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

5. Simplified 50.7%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. +-lowering-+.f64 50.7%

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

7. Applied egg-rr 50.7%

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

8. Final simplification 50.7%

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

Alternative 11: 50.1% 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. /-lowering-/.f64 N/A

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

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

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

   3. +-commutative N/A

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

   4. +-lowering-+.f64 50.7%

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

5. Simplified 50.7%

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

6. Taylor expanded in e around 0

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

   1. *-lowering-*.f64 49.8%

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

8. Simplified 49.8%

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

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

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

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

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

   3. +-commutative N/A

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

   4. +-lowering-+.f64 50.7%

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

5. Simplified 50.7%

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

6. Taylor expanded in e around inf

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

8. Simplified 4.2%

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

Reproduce

herbie shell --seed 2024164 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))