Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 7.9s

Alternatives: 11

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

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

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(Float64(e / 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[(N[(e / N[(1.0 + N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in v around inf 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 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. Step-by-step derivation

   1. associate-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in v around 0 99.0%

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

   1. +-commutative 99.0%

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

6. Simplified 99.0%

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

7. Final simplification 99.0%

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

Alternative 3: 98.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/l* 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 98.8%

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

   2. associate-/r/ 99.6%

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

   3. clear-num 99.8%

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

   4. +-commutative 99.8%

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

   5. fma-udef 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in v around inf 99.8%

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

7. Taylor expanded in v around 0 99.0%

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

8. Final simplification 99.0%

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

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. Step-by-step derivation

   1. associate-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in e around 0 98.2%

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

5. Final simplification 98.2%

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

Alternative 5: 53.1% 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* 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in v around 0 57.7%

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

5. Taylor expanded in e around 0 57.7%

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

6. Final simplification 57.7%

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

Alternative 6: 52.7% accurate, 13.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e / N[(N[(N[(e / v), $MachinePrecision] + N[(1.0 / v), $MachinePrecision]), $MachinePrecision] + N[(-0.3333333333333333 * N[(e * 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* 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in v around 0 57.7%

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

5. Taylor expanded in e around inf 57.5%

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

6. Final simplification 57.5%

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

Alternative 7: 51.6% accurate, 29.9× speedup

Math

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

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(Float64(e * v) * Float64(1.0 - e))
end

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(e * v), $MachinePrecision] * N[(1.0 - 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/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in v around 0 57.0%

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

   1. associate-/l* 56.9%

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

   2. associate-/r/ 57.0%

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

   3. +-commutative 57.0%

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

6. Simplified 57.0%

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

7. Taylor expanded in e around 0 56.8%

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

   1. mul-1-neg 56.8%

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

   2. unsub-neg 56.8%

      \[\leadsto \color{blue}{v \cdot e - v \cdot {e}^{2}} \]

   3. *-commutative 56.8%

      \[\leadsto \color{blue}{e \cdot v} - v \cdot {e}^{2} \]

   4. *-commutative 56.8%

      \[\leadsto e \cdot v - \color{blue}{{e}^{2} \cdot v} \]

   5. unpow2 56.8%

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

   6. associate-*l* 56.8%

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

9. Simplified 56.8%

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

    1. *-un-lft-identity 56.8%

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

    2. distribute-rgt-out-- 56.8%

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

    3. *-commutative 56.8%

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

11. Applied egg-rr 56.8%

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

12. Final simplification 56.8%

    \[\leadsto \left(e \cdot v\right) \cdot \left(1 - e\right) \]

Alternative 8: 52.0% 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. Step-by-step derivation

   1. associate-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in v around 0 57.0%

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

   1. associate-/l* 56.9%

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

   2. associate-/r/ 57.0%

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

   3. +-commutative 57.0%

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

6. Simplified 57.0%

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

7. Final simplification 57.0%

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

Alternative 9: 52.0% 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/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in v around 0 57.0%

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

5. Final simplification 57.0%

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

Alternative 10: 51.2% 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/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in v around 0 57.0%

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

   1. associate-/l* 56.9%

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

   2. associate-/r/ 57.0%

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

   3. +-commutative 57.0%

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

6. Simplified 57.0%

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

7. Taylor expanded in e around 0 56.2%

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

8. Final simplification 56.2%

   \[\leadsto e \cdot v \]

Alternative 11: 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* 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in v around 0 57.7%

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

5. Taylor expanded in e around inf 5.9%

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

   1. *-commutative 5.9%

      \[\leadsto \frac{1}{\frac{1}{v} + \color{blue}{v \cdot -0.3333333333333333}} \]

7. Simplified 5.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{v} + v \cdot -0.3333333333333333}} \]

8. Taylor expanded in v around 0 4.9%

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

9. Final simplification 4.9%

   \[\leadsto v \]

Reproduce

herbie shell --seed 2023252 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))