Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 6.7s

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. Final simplification 99.8%

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

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. Step-by-step derivation

   1. associate-/l* 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 3: 98.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/l* 99.7%

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

3. Simplified 99.7%

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

   1. add-sqr-sqrt 99.3%

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

   2. div-inv 99.3%

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

   3. times-frac 99.3%

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

   4. +-commutative 99.3%

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

   5. fma-def 99.3%

      \[\leadsto \frac{\sqrt{e}}{\color{blue}{\mathsf{fma}\left(e, \cos v, 1\right)}} \cdot \frac{\sqrt{e}}{\frac{1}{\sin v}} \]

5. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\frac{\sqrt{e}}{\mathsf{fma}\left(e, \cos v, 1\right)} \cdot \frac{\sqrt{e}}{\frac{1}{\sin v}}} \]

6. Taylor expanded in v around 0 98.8%

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

   1. associate-*r/ 98.8%

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

   2. *-rgt-identity 98.8%

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

   3. +-commutative 98.8%

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

8. Simplified 98.8%

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

9. Taylor expanded in v around inf 99.3%

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

    1. associate-/l* 99.1%

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

11. Simplified 99.1%

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

12. Final simplification 99.1%

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

Alternative 4: 97.6% 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. Step-by-step derivation

   1. associate-/l* 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in e around 0 98.4%

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

5. Final simplification 98.4%

   \[\leadsto e \cdot \sin v \]

Alternative 5: 52.7% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/l* 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in v around 0 54.1%

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

5. Taylor expanded in e around 0 54.1%

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

6. Final simplification 54.1%

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

Alternative 6: 52.4% accurate, 13.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/l* 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in v around 0 54.1%

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

5. Taylor expanded in e around inf 53.8%

   \[\leadsto \frac{e}{\color{blue}{-0.3333333333333333 \cdot \left(v \cdot e\right)} + \left(\frac{e}{v} + \frac{1}{v}\right)} \]
6. Step-by-step derivation

   1. associate-*r* 53.8%

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

   2. *-commutative 53.8%

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

   3. associate-*l* 53.8%

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

7. Simplified 53.8%

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

8. Final simplification 53.8%

   \[\leadsto \frac{e}{\left(\frac{e}{v} + \frac{1}{v}\right) + v \cdot \left(e \cdot -0.3333333333333333\right)} \]

Alternative 7: 51.7% accurate, 23.2× speedup

Math

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

FPCore

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

C

double code(double e, double v) {
	return 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 = e / ((1.0d0 / v) + (v * 0.16666666666666666d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/l* 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in v around 0 54.1%

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

5. Taylor expanded in e around 0 53.3%

   \[\leadsto \color{blue}{\frac{e}{0.16666666666666666 \cdot v + \frac{1}{v}}} \]

6. Final simplification 53.3%

   \[\leadsto \frac{e}{\frac{1}{v} + v \cdot 0.16666666666666666} \]

Alternative 8: 51.6% 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. Step-by-step derivation

   1. associate-/l* 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in v around 0 52.8%

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

   1. +-commutative 52.8%

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

6. Simplified 52.8%

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

7. Final simplification 52.8%

   \[\leadsto \frac{e}{\frac{e + 1}{v}} \]

Alternative 9: 51.6% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/l* 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in v around 0 52.9%

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

   1. associate-/l* 52.8%

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

   2. +-commutative 52.8%

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

6. Simplified 52.8%

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

7. Final simplification 52.8%

   \[\leadsto \frac{v}{\frac{e + 1}{e}} \]

Alternative 10: 51.7% 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. Step-by-step derivation

   1. associate-/l* 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in v around 0 52.9%

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

5. Final simplification 52.9%

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

Alternative 11: 50.6% 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. Step-by-step derivation

   1. associate-/l* 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in v around 0 52.8%

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

   1. +-commutative 52.8%

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

6. Simplified 52.8%

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

7. Taylor expanded in e around 0 52.0%

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

8. Final simplification 52.0%

   \[\leadsto e \cdot v \]

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. Step-by-step derivation

   1. associate-/l* 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in v around 0 52.8%

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

   1. +-commutative 52.8%

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

6. Simplified 52.8%

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

7. Taylor expanded in e around inf 4.5%

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

8. Final simplification 4.5%

   \[\leadsto v \]

Reproduce

herbie shell --seed 2023230 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))