Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 8.7s

Alternatives: 9

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: 98.9% accurate, 2.0× speedup

Math

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

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(Float64(e * 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[(N[(e * N[Sin[v], $MachinePrecision]), $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

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 98.9%

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

5. Simplified 98.9%

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

Alternative 4: 98.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in e around 0

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

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

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

   2. sin-lowering-sin.f64 98.7%

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

5. Simplified 98.7%

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

Alternative 5: 52.8% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. 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

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

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

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

   2. associate-+r+ N/A

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

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

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

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

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

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

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

   6. unpow2 N/A

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

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

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

   8. sub-neg N/A

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

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

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

   10. *-commutative N/A

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

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

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

   12. distribute-lft-neg-in N/A

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

   13. metadata-eval N/A

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

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

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

   15. +-lowering-+.f64 54.2%

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

7. Simplified 54.2%

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

8. Final simplification 54.2%

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

Alternative 6: 52.0% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \frac{e \cdot v}{1 + \left(v \cdot v\right) \cdot \left(0.16666666666666666 + \left(v \cdot v\right) \cdot 0.019444444444444445\right)} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (/
  (* e v)
  (+
   1.0
   (* (* v v) (+ 0.16666666666666666 (* (* v v) 0.019444444444444445))))))

C

double code(double e, double v) {
	return (e * v) / (1.0 + ((v * v) * (0.16666666666666666 + ((v * v) * 0.019444444444444445))));
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = (e * v) / (1.0d0 + ((v * v) * (0.16666666666666666d0 + ((v * v) * 0.019444444444444445d0))))
end function

Java

public static double code(double e, double v) {
	return (e * v) / (1.0 + ((v * v) * (0.16666666666666666 + ((v * v) * 0.019444444444444445))));
}

Python

def code(e, v):
	return (e * v) / (1.0 + ((v * v) * (0.16666666666666666 + ((v * v) * 0.019444444444444445))))

Julia

function code(e, v)
	return Float64(Float64(e * v) / Float64(1.0 + Float64(Float64(v * v) * Float64(0.16666666666666666 + Float64(Float64(v * v) * 0.019444444444444445)))))
end

MATLAB

function tmp = code(e, v)
	tmp = (e * v) / (1.0 + ((v * v) * (0.16666666666666666 + ((v * v) * 0.019444444444444445))));
end

Wolfram

code[e_, v_] := N[(N[(e * v), $MachinePrecision] / N[(1.0 + N[(N[(v * v), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(v * v), $MachinePrecision] * 0.019444444444444445), $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

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

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

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

7. Simplified 54.1%

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

8. Taylor expanded in e around 0

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + {v}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {v}^{2}\right)}} \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(\left(e \cdot v\right), \color{blue}{\left(1 + {v}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {v}^{2}\right)\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(v \cdot e\right), \left(\color{blue}{1} + {v}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {v}^{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, e\right), \left(\color{blue}{1} + {v}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {v}^{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, e\right), \mathsf{+.f64}\left(1, \color{blue}{\left({v}^{2} \cdot \left(\frac{1}{6} + \frac{7}{360} \cdot {v}^{2}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, e\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({v}^{2}\right), \color{blue}{\left(\frac{1}{6} + \frac{7}{360} \cdot {v}^{2}\right)}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, e\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(v \cdot v\right), \left(\color{blue}{\frac{1}{6}} + \frac{7}{360} \cdot {v}^{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, e\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\color{blue}{\frac{1}{6}} + \frac{7}{360} \cdot {v}^{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, e\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{7}{360} \cdot {v}^{2}\right)}\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, e\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({v}^{2} \cdot \color{blue}{\frac{7}{360}}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, e\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({v}^{2}\right), \color{blue}{\frac{7}{360}}\right)\right)\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, e\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(v \cdot v\right), \frac{7}{360}\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 54.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(v, e\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \frac{7}{360}\right)\right)\right)\right)\right) \]

10. Simplified 54.0%

    \[\leadsto \color{blue}{\frac{v \cdot e}{1 + \left(v \cdot v\right) \cdot \left(0.16666666666666666 + \left(v \cdot v\right) \cdot 0.019444444444444445\right)}} \]

11. Final simplification 54.0%

    \[\leadsto \frac{e \cdot v}{1 + \left(v \cdot v\right) \cdot \left(0.16666666666666666 + \left(v \cdot v\right) \cdot 0.019444444444444445\right)} \]
12. Add Preprocessing

Alternative 7: 51.3% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around 0

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

   1. /-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 53.0%

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

5. Simplified 53.0%

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

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. *-lft-identity N/A

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

   3. metadata-eval N/A

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

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

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

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

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

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

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

   7. metadata-eval N/A

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

   8. *-lft-identity N/A

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

   9. +-commutative N/A

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

   10. mul-1-neg N/A

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

   11. unsub-neg N/A

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

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

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

   13. *-commutative N/A

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

   14. *-lowering-*.f64 53.0%

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

8. Simplified 53.0%

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

9. Final simplification 53.0%

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

Alternative 8: 50.8% 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 53.0%

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

5. Simplified 53.0%

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

6. Taylor expanded in e around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 52.8%

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

8. Simplified 52.8%

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

9. Final simplification 52.8%

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

Alternative 9: 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 53.0%

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

5. Simplified 53.0%

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

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 2024170 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))