Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 10.0s

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: 99.6% 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. 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 3: 98.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[Sin[v], $MachinePrecision] * N[(e / 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 99.0%

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

5. Simplified 99.0%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

   6. +-lowering-+.f64 99.0%

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

7. Applied egg-rr 99.0%

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

Alternative 4: 98.2% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (e v) :precision binary64 (* e (* (sin v) (- 1.0 e))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e * N[(N[Sin[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. Step-by-step derivation

   1. associate-/l* N/A

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

   2. clear-num N/A

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

   3. un-div-inv 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. *-lowering-*.f64 N/A

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

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

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

   9. 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. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 98.8%

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

7. Simplified 98.8%

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

8. Taylor expanded in e around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

   4. sub-neg N/A

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

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

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

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

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

   9. distribute-rgt-out N/A

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

   10. +-commutative N/A

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

   11. neg-mul-1 N/A

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

   12. sub-neg N/A

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

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

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

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

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

   15. --lowering--.f64 98.6%

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

10. Simplified 98.6%

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

Alternative 5: 97.7% 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 97.9%

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

5. Simplified 97.9%

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

Alternative 6: 53.1% accurate, 13.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. clear-num N/A

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

   3. un-div-inv 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. *-lowering-*.f64 N/A

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

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

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

   9. 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. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 98.8%

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

7. Simplified 98.8%

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

8. Taylor expanded in v around 0

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

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

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt1-in N/A

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

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

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

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

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

   7. *-commutative N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

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

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

   12. *-commutative N/A

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

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

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

   14. +-lowering-+.f64 55.3%

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

10. Simplified 55.3%

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

    1. associate-/r/ N/A

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

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

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

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

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

    4. +-commutative N/A

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

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

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

    6. +-commutative N/A

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

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

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

    8. associate-*r* N/A

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

    9. *-commutative N/A

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

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

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

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

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

    12. +-commutative N/A

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

    13. +-lowering-+.f64 55.3%

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

12. Applied egg-rr 55.3%

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

13. Final simplification 55.3%

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

Alternative 7: 52.0% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(e * v), $MachinePrecision] / N[(1.0 + N[(0.16666666666666666 * N[(v * 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. Step-by-step derivation

   1. associate-/l* N/A

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

   2. clear-num N/A

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

   3. un-div-inv 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. *-lowering-*.f64 N/A

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

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

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

   9. 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. Taylor expanded in v around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 98.8%

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

7. Simplified 98.8%

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

8. Taylor expanded in v around 0

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

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

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

   2. associate-+r+ N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt1-in N/A

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

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

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

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

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

   7. *-commutative N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

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

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

   12. *-commutative N/A

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

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

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

   14. +-lowering-+.f64 55.3%

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

10. Simplified 55.3%

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

11. Taylor expanded in e around 0

    \[\leadsto \color{blue}{\frac{e \cdot v}{1 + \frac{1}{6} \cdot {v}^{2}}} \]
12. 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. *-commutative N/A

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

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

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

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

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

    5. *-commutative N/A

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

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

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

    7. unpow2 N/A

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

    8. *-lowering-*.f64 54.2%

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

13. Simplified 54.2%

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

14. Final simplification 54.2%

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

Alternative 8: 51.9% accurate, 29.9× speedup

Math

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

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(Float64(e * v) / Float64(e + 1.0))
end

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(e * v), $MachinePrecision] / N[(e + 1.0), $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 54.2%

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

5. Simplified 54.2%

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

Alternative 9: 51.8% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(v / N[(1.0 + 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. Taylor expanded in v around 0

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

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

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

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

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

   3. /-lowering-/.f64 54.1%

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

8. Simplified 54.1%

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

Alternative 10: 51.8% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. clear-num N/A

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

   3. un-div-inv 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. *-lowering-*.f64 N/A

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

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

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

   9. 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. Taylor expanded in v around 0

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

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

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

   2. +-commutative N/A

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

   3. +-lowering-+.f64 54.1%

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

7. Simplified 54.1%

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

Alternative 11: 51.4% accurate, 29.9× speedup

Math

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

FPCore

(FPCore (e v) :precision binary64 (* e (* v (- 1.0 e))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e * N[(v * 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 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 54.2%

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

5. Simplified 54.2%

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

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. *-lft-identity N/A

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

   11. associate-*r* N/A

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

   12. distribute-rgt-in N/A

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

   13. neg-mul-1 N/A

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

   14. sub-neg N/A

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

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

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

   16. --lowering--.f64 53.8%

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

8. Simplified 53.8%

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

Alternative 12: 50.9% 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 54.2%

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

5. Simplified 54.2%

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

6. Taylor expanded in e around 0

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

   1. *-lowering-*.f64 53.1%

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

8. Simplified 53.1%

   \[\leadsto \color{blue}{e \cdot v} \]
9. 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. /-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 54.2%

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

5. Simplified 54.2%

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

6. Taylor expanded in e around inf

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

8. Simplified 4.6%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))