Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 9.2s

Alternatives: 15

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{e \cdot \sin v}{\mathsf{fma}\left(\cos v, e, 1\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. fma-define N/A

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

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

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

   5. cos-lowering-cos.f64 99.8%

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

4. Applied egg-rr 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-/l* N/A

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

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

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

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

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

   5. rgt-mult-inverse N/A

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

   6. distribute-lft-in N/A

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

   7. +-commutative N/A

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

   8. associate-/r* N/A

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

   9. *-rgt-identity N/A

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

   10. associate-*r/ N/A

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

   11. rgt-mult-inverse N/A

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

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

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

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

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

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

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

   15. /-lowering-/.f64 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 4: 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. *-commutative N/A

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

   2. associate-/l* N/A

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

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

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

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

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

   5. rgt-mult-inverse N/A

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

   6. distribute-lft-in N/A

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

   7. +-commutative N/A

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

   8. associate-/r* N/A

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

   9. *-rgt-identity N/A

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

   10. associate-*r/ N/A

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

   11. rgt-mult-inverse N/A

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

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

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

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

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

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

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

   15. /-lowering-/.f64 99.7%

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

5. Simplified 99.7%

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

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 5: 98.7% 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 98.5%

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

5. Simplified 98.5%

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

Alternative 6: 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. 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 98.5%

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

5. Simplified 98.5%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

   6. +-lowering-+.f64 98.5%

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

7. Applied egg-rr 98.5%

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

8. Final simplification 98.5%

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

Alternative 7: 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. 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 97.8%

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

5. Simplified 97.8%

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

Alternative 8: 51.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e \cdot 0.041666666666666664 - \left(0.008333333333333333 + e \cdot 0.008333333333333333\right)\\ t_1 := 0.16666666666666666 + e \cdot 0.16666666666666666\\ t_2 := e \cdot -0.5 + t\_1\\ t_3 := t\_2 \cdot -0.16666666666666666\\ \frac{e}{\frac{\left(e + 1\right) + v \cdot \left(v \cdot \left(e \cdot -0.5 + \left(t\_1 + \left(v \cdot v\right) \cdot \left(\left(t\_0 - t\_3\right) + v \cdot \left(v \cdot \left(\left(e \cdot -0.001388888888888889 - \left(-0.0001984126984126984 + \left(e \cdot -0.0001984126984126984 + 0.008333333333333333 \cdot t\_2\right)\right)\right) + -0.16666666666666666 \cdot \left(t\_3 - t\_0\right)\right)\right)\right)\right)\right)\right)}{v}} \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (let* ((t_0
         (-
          (* e 0.041666666666666664)
          (+ 0.008333333333333333 (* e 0.008333333333333333))))
        (t_1 (+ 0.16666666666666666 (* e 0.16666666666666666)))
        (t_2 (+ (* e -0.5) t_1))
        (t_3 (* t_2 -0.16666666666666666)))
   (/
    e
    (/
     (+
      (+ e 1.0)
      (*
       v
       (*
        v
        (+
         (* e -0.5)
         (+
          t_1
          (*
           (* v v)
           (+
            (- t_0 t_3)
            (*
             v
             (*
              v
              (+
               (-
                (* e -0.001388888888888889)
                (+
                 -0.0001984126984126984
                 (+
                  (* e -0.0001984126984126984)
                  (* 0.008333333333333333 t_2))))
               (* -0.16666666666666666 (- t_3 t_0))))))))))))
     v))))

C

double code(double e, double v) {
	double t_0 = (e * 0.041666666666666664) - (0.008333333333333333 + (e * 0.008333333333333333));
	double t_1 = 0.16666666666666666 + (e * 0.16666666666666666);
	double t_2 = (e * -0.5) + t_1;
	double t_3 = t_2 * -0.16666666666666666;
	return e / (((e + 1.0) + (v * (v * ((e * -0.5) + (t_1 + ((v * v) * ((t_0 - t_3) + (v * (v * (((e * -0.001388888888888889) - (-0.0001984126984126984 + ((e * -0.0001984126984126984) + (0.008333333333333333 * t_2)))) + (-0.16666666666666666 * (t_3 - t_0)))))))))))) / v);
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    t_0 = (e * 0.041666666666666664d0) - (0.008333333333333333d0 + (e * 0.008333333333333333d0))
    t_1 = 0.16666666666666666d0 + (e * 0.16666666666666666d0)
    t_2 = (e * (-0.5d0)) + t_1
    t_3 = t_2 * (-0.16666666666666666d0)
    code = e / (((e + 1.0d0) + (v * (v * ((e * (-0.5d0)) + (t_1 + ((v * v) * ((t_0 - t_3) + (v * (v * (((e * (-0.001388888888888889d0)) - ((-0.0001984126984126984d0) + ((e * (-0.0001984126984126984d0)) + (0.008333333333333333d0 * t_2)))) + ((-0.16666666666666666d0) * (t_3 - t_0)))))))))))) / v)
end function

Java

public static double code(double e, double v) {
	double t_0 = (e * 0.041666666666666664) - (0.008333333333333333 + (e * 0.008333333333333333));
	double t_1 = 0.16666666666666666 + (e * 0.16666666666666666);
	double t_2 = (e * -0.5) + t_1;
	double t_3 = t_2 * -0.16666666666666666;
	return e / (((e + 1.0) + (v * (v * ((e * -0.5) + (t_1 + ((v * v) * ((t_0 - t_3) + (v * (v * (((e * -0.001388888888888889) - (-0.0001984126984126984 + ((e * -0.0001984126984126984) + (0.008333333333333333 * t_2)))) + (-0.16666666666666666 * (t_3 - t_0)))))))))))) / v);
}

Python

def code(e, v):
	t_0 = (e * 0.041666666666666664) - (0.008333333333333333 + (e * 0.008333333333333333))
	t_1 = 0.16666666666666666 + (e * 0.16666666666666666)
	t_2 = (e * -0.5) + t_1
	t_3 = t_2 * -0.16666666666666666
	return e / (((e + 1.0) + (v * (v * ((e * -0.5) + (t_1 + ((v * v) * ((t_0 - t_3) + (v * (v * (((e * -0.001388888888888889) - (-0.0001984126984126984 + ((e * -0.0001984126984126984) + (0.008333333333333333 * t_2)))) + (-0.16666666666666666 * (t_3 - t_0)))))))))))) / v)

Julia

function code(e, v)
	t_0 = Float64(Float64(e * 0.041666666666666664) - Float64(0.008333333333333333 + Float64(e * 0.008333333333333333)))
	t_1 = Float64(0.16666666666666666 + Float64(e * 0.16666666666666666))
	t_2 = Float64(Float64(e * -0.5) + t_1)
	t_3 = Float64(t_2 * -0.16666666666666666)
	return Float64(e / Float64(Float64(Float64(e + 1.0) + Float64(v * Float64(v * Float64(Float64(e * -0.5) + Float64(t_1 + Float64(Float64(v * v) * Float64(Float64(t_0 - t_3) + Float64(v * Float64(v * Float64(Float64(Float64(e * -0.001388888888888889) - Float64(-0.0001984126984126984 + Float64(Float64(e * -0.0001984126984126984) + Float64(0.008333333333333333 * t_2)))) + Float64(-0.16666666666666666 * Float64(t_3 - t_0)))))))))))) / v))
end

MATLAB

function tmp = code(e, v)
	t_0 = (e * 0.041666666666666664) - (0.008333333333333333 + (e * 0.008333333333333333));
	t_1 = 0.16666666666666666 + (e * 0.16666666666666666);
	t_2 = (e * -0.5) + t_1;
	t_3 = t_2 * -0.16666666666666666;
	tmp = e / (((e + 1.0) + (v * (v * ((e * -0.5) + (t_1 + ((v * v) * ((t_0 - t_3) + (v * (v * (((e * -0.001388888888888889) - (-0.0001984126984126984 + ((e * -0.0001984126984126984) + (0.008333333333333333 * t_2)))) + (-0.16666666666666666 * (t_3 - t_0)))))))))))) / v);
end

Wolfram

code[e_, v_] := Block[{t$95$0 = N[(N[(e * 0.041666666666666664), $MachinePrecision] - N[(0.008333333333333333 + N[(e * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.16666666666666666 + N[(e * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(e * -0.5), $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * -0.16666666666666666), $MachinePrecision]}, N[(e / N[(N[(N[(e + 1.0), $MachinePrecision] + N[(v * N[(v * N[(N[(e * -0.5), $MachinePrecision] + N[(t$95$1 + N[(N[(v * v), $MachinePrecision] * N[(N[(t$95$0 - t$95$3), $MachinePrecision] + N[(v * N[(v * N[(N[(N[(e * -0.001388888888888889), $MachinePrecision] - N[(-0.0001984126984126984 + N[(N[(e * -0.0001984126984126984), $MachinePrecision] + N[(0.008333333333333333 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(t$95$3 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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

   \[\leadsto \mathsf{/.f64}\left(e, \color{blue}{\left(\frac{1 + \left(e + {v}^{2} \cdot \left(\left(\frac{-1}{2} \cdot e + {v}^{2} \cdot \left(\left(\frac{1}{24} \cdot e + {v}^{2} \cdot \left(\frac{-1}{720} \cdot e - \left(\frac{-1}{6} \cdot \left(\frac{1}{24} \cdot e - \left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right) + \frac{1}{120} \cdot \left(1 + e\right)\right)\right) + \left(\frac{-1}{5040} \cdot \left(1 + e\right) + \frac{1}{120} \cdot \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right)\right)\right)\right)\right) - \left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right) + \frac{1}{120} \cdot \left(1 + e\right)\right)\right)\right) - \frac{-1}{6} \cdot \left(1 + e\right)\right)\right)}{v}\right)}\right) \]

7. Simplified 52.0%

   \[\leadsto \frac{e}{\color{blue}{\frac{\left(e + 1\right) + v \cdot \left(v \cdot \left(e \cdot -0.5 + \left(\left(v \cdot v\right) \cdot \left(v \cdot \left(v \cdot \left(\left(e \cdot -0.001388888888888889 - \left(-0.0001984126984126984 + \left(-0.0001984126984126984 \cdot e + 0.008333333333333333 \cdot \left(e \cdot -0.5 + \left(0.16666666666666666 + e \cdot 0.16666666666666666\right)\right)\right)\right)\right) - -0.16666666666666666 \cdot \left(\left(e \cdot 0.041666666666666664 - \left(0.008333333333333333 + e \cdot 0.008333333333333333\right)\right) - -0.16666666666666666 \cdot \left(e \cdot -0.5 + \left(0.16666666666666666 + e \cdot 0.16666666666666666\right)\right)\right)\right)\right) + \left(\left(e \cdot 0.041666666666666664 - \left(0.008333333333333333 + e \cdot 0.008333333333333333\right)\right) - -0.16666666666666666 \cdot \left(e \cdot -0.5 + \left(0.16666666666666666 + e \cdot 0.16666666666666666\right)\right)\right)\right) + \left(0.16666666666666666 + e \cdot 0.16666666666666666\right)\right)\right)\right)}{v}}} \]

8. Final simplification 52.0%

   \[\leadsto \frac{e}{\frac{\left(e + 1\right) + v \cdot \left(v \cdot \left(e \cdot -0.5 + \left(\left(0.16666666666666666 + e \cdot 0.16666666666666666\right) + \left(v \cdot v\right) \cdot \left(\left(\left(e \cdot 0.041666666666666664 - \left(0.008333333333333333 + e \cdot 0.008333333333333333\right)\right) - \left(e \cdot -0.5 + \left(0.16666666666666666 + e \cdot 0.16666666666666666\right)\right) \cdot -0.16666666666666666\right) + v \cdot \left(v \cdot \left(\left(e \cdot -0.001388888888888889 - \left(-0.0001984126984126984 + \left(e \cdot -0.0001984126984126984 + 0.008333333333333333 \cdot \left(e \cdot -0.5 + \left(0.16666666666666666 + e \cdot 0.16666666666666666\right)\right)\right)\right)\right) + -0.16666666666666666 \cdot \left(\left(e \cdot -0.5 + \left(0.16666666666666666 + e \cdot 0.16666666666666666\right)\right) \cdot -0.16666666666666666 - \left(e \cdot 0.041666666666666664 - \left(0.008333333333333333 + e \cdot 0.008333333333333333\right)\right)\right)\right)\right)\right)\right)\right)\right)}{v}} \]
9. Add Preprocessing

Alternative 9: 51.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e \cdot -0.5 + \left(0.16666666666666666 + e \cdot 0.16666666666666666\right)\\ \frac{e}{\frac{\left(e + \left(v \cdot v\right) \cdot \left(t\_0 + \left(v \cdot v\right) \cdot \left(\left(e \cdot 0.041666666666666664 - \left(0.008333333333333333 + e \cdot 0.008333333333333333\right)\right) - t\_0 \cdot -0.16666666666666666\right)\right)\right) + 1}{v}} \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (let* ((t_0 (+ (* e -0.5) (+ 0.16666666666666666 (* e 0.16666666666666666)))))
   (/
    e
    (/
     (+
      (+
       e
       (*
        (* v v)
        (+
         t_0
         (*
          (* v v)
          (-
           (-
            (* e 0.041666666666666664)
            (+ 0.008333333333333333 (* e 0.008333333333333333)))
           (* t_0 -0.16666666666666666))))))
      1.0)
     v))))

C

double code(double e, double v) {
	double t_0 = (e * -0.5) + (0.16666666666666666 + (e * 0.16666666666666666));
	return e / (((e + ((v * v) * (t_0 + ((v * v) * (((e * 0.041666666666666664) - (0.008333333333333333 + (e * 0.008333333333333333))) - (t_0 * -0.16666666666666666)))))) + 1.0) / v);
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    real(8) :: t_0
    t_0 = (e * (-0.5d0)) + (0.16666666666666666d0 + (e * 0.16666666666666666d0))
    code = e / (((e + ((v * v) * (t_0 + ((v * v) * (((e * 0.041666666666666664d0) - (0.008333333333333333d0 + (e * 0.008333333333333333d0))) - (t_0 * (-0.16666666666666666d0))))))) + 1.0d0) / v)
end function

Java

public static double code(double e, double v) {
	double t_0 = (e * -0.5) + (0.16666666666666666 + (e * 0.16666666666666666));
	return e / (((e + ((v * v) * (t_0 + ((v * v) * (((e * 0.041666666666666664) - (0.008333333333333333 + (e * 0.008333333333333333))) - (t_0 * -0.16666666666666666)))))) + 1.0) / v);
}

Python

def code(e, v):
	t_0 = (e * -0.5) + (0.16666666666666666 + (e * 0.16666666666666666))
	return e / (((e + ((v * v) * (t_0 + ((v * v) * (((e * 0.041666666666666664) - (0.008333333333333333 + (e * 0.008333333333333333))) - (t_0 * -0.16666666666666666)))))) + 1.0) / v)

Julia

function code(e, v)
	t_0 = Float64(Float64(e * -0.5) + Float64(0.16666666666666666 + Float64(e * 0.16666666666666666)))
	return Float64(e / Float64(Float64(Float64(e + Float64(Float64(v * v) * Float64(t_0 + Float64(Float64(v * v) * Float64(Float64(Float64(e * 0.041666666666666664) - Float64(0.008333333333333333 + Float64(e * 0.008333333333333333))) - Float64(t_0 * -0.16666666666666666)))))) + 1.0) / v))
end

MATLAB

function tmp = code(e, v)
	t_0 = (e * -0.5) + (0.16666666666666666 + (e * 0.16666666666666666));
	tmp = e / (((e + ((v * v) * (t_0 + ((v * v) * (((e * 0.041666666666666664) - (0.008333333333333333 + (e * 0.008333333333333333))) - (t_0 * -0.16666666666666666)))))) + 1.0) / v);
end

Wolfram

code[e_, v_] := Block[{t$95$0 = N[(N[(e * -0.5), $MachinePrecision] + N[(0.16666666666666666 + N[(e * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(e / N[(N[(N[(e + N[(N[(v * v), $MachinePrecision] * N[(t$95$0 + N[(N[(v * v), $MachinePrecision] * N[(N[(N[(e * 0.041666666666666664), $MachinePrecision] - N[(0.008333333333333333 + N[(e * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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

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

7. Simplified 51.9%

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

8. Final simplification 51.9%

   \[\leadsto \frac{e}{\frac{\left(e + \left(v \cdot v\right) \cdot \left(\left(e \cdot -0.5 + \left(0.16666666666666666 + e \cdot 0.16666666666666666\right)\right) + \left(v \cdot v\right) \cdot \left(\left(e \cdot 0.041666666666666664 - \left(0.008333333333333333 + e \cdot 0.008333333333333333\right)\right) - \left(e \cdot -0.5 + \left(0.16666666666666666 + e \cdot 0.16666666666666666\right)\right) \cdot -0.16666666666666666\right)\right)\right) + 1}{v}} \]
9. Add Preprocessing

Alternative 10: 50.4% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 3.1:\\ \;\;\;\;v \cdot \left(\frac{e}{e + 1} + \left(v \cdot v\right) \cdot \left(e \cdot \left(-0.16666666666666666 + \left(v \cdot v\right) \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - e \cdot \left(e \cdot v\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (if (<= v 3.1)
   (*
    v
    (+
     (/ e (+ e 1.0))
     (*
      (* v v)
      (* e (+ -0.16666666666666666 (* (* v v) 0.008333333333333333))))))
   (- 0.0 (* e (* e v)))))

C

double code(double e, double v) {
	double tmp;
	if (v <= 3.1) {
		tmp = v * ((e / (e + 1.0)) + ((v * v) * (e * (-0.16666666666666666 + ((v * v) * 0.008333333333333333)))));
	} else {
		tmp = 0.0 - (e * (e * v));
	}
	return tmp;
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    real(8) :: tmp
    if (v <= 3.1d0) then
        tmp = v * ((e / (e + 1.0d0)) + ((v * v) * (e * ((-0.16666666666666666d0) + ((v * v) * 0.008333333333333333d0)))))
    else
        tmp = 0.0d0 - (e * (e * v))
    end if
    code = tmp
end function

Java

public static double code(double e, double v) {
	double tmp;
	if (v <= 3.1) {
		tmp = v * ((e / (e + 1.0)) + ((v * v) * (e * (-0.16666666666666666 + ((v * v) * 0.008333333333333333)))));
	} else {
		tmp = 0.0 - (e * (e * v));
	}
	return tmp;
}

Python

def code(e, v):
	tmp = 0
	if v <= 3.1:
		tmp = v * ((e / (e + 1.0)) + ((v * v) * (e * (-0.16666666666666666 + ((v * v) * 0.008333333333333333)))))
	else:
		tmp = 0.0 - (e * (e * v))
	return tmp

Julia

function code(e, v)
	tmp = 0.0
	if (v <= 3.1)
		tmp = Float64(v * Float64(Float64(e / Float64(e + 1.0)) + Float64(Float64(v * v) * Float64(e * Float64(-0.16666666666666666 + Float64(Float64(v * v) * 0.008333333333333333))))));
	else
		tmp = Float64(0.0 - Float64(e * Float64(e * v)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(e, v)
	tmp = 0.0;
	if (v <= 3.1)
		tmp = v * ((e / (e + 1.0)) + ((v * v) * (e * (-0.16666666666666666 + ((v * v) * 0.008333333333333333)))));
	else
		tmp = 0.0 - (e * (e * v));
	end
	tmp_2 = tmp;
end

Wolfram

code[e_, v_] := If[LessEqual[v, 3.1], N[(v * N[(N[(e / N[(e + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(v * v), $MachinePrecision] * N[(e * N[(-0.16666666666666666 + N[(N[(v * v), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[(e * N[(e * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 3.10000000000000009

   1. Initial program 99.9%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 68.1%

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

   6. Taylor expanded in e around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 68.1%

         \[\leadsto \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(\mathsf{/.f64}\left(e, \mathsf{+.f64}\left(e, 1\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(e, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \frac{1}{120}\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

   8. Simplified 68.1%

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

   if 3.10000000000000009 < v

   1. Initial program 99.5%

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

   3. Taylor expanded in v around 0

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

   5. Simplified 2.3%

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

   6. Taylor expanded in e around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 2.3%

         \[\leadsto \mathsf{*.f64}\left(v, \mathsf{+.f64}\left(\mathsf{/.f64}\left(e, \mathsf{+.f64}\left(e, 1\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{*.f64}\left(e, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \frac{1}{120}\right), \frac{-1}{6}\right)\right)\right)\right)\right) \]

   8. Simplified 2.3%

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

   9. Taylor expanded in e around 0

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

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

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. associate-*r* N/A

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

       5. distribute-rgt-out N/A

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

       6. associate-+r+ N/A

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

       7. +-commutative N/A

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

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

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

       9. associate-+r+ N/A

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

       10. +-commutative N/A

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

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

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

   11. Simplified 2.3%

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

   12. Taylor expanded in e around inf

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

       1. *-commutative N/A

          \[\leadsto \left({e}^{2} \cdot v\right) \cdot \color{blue}{-1} \]

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

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

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

       7. mul-1-neg N/A

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

       8. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(e, \left(0 - \color{blue}{e \cdot v}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(e, \mathsf{\_.f64}\left(0, \color{blue}{\left(e \cdot v\right)}\right)\right) \]

       10. *-lowering-*.f64 5.8%

           \[\leadsto \mathsf{*.f64}\left(e, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(e, \color{blue}{v}\right)\right)\right) \]

   14. Simplified 5.8%

       \[\leadsto \color{blue}{e \cdot \left(0 - e \cdot v\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 3.1:\\ \;\;\;\;v \cdot \left(\frac{e}{e + 1} + \left(v \cdot v\right) \cdot \left(e \cdot \left(-0.16666666666666666 + \left(v \cdot v\right) \cdot 0.008333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0 - e \cdot \left(e \cdot v\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.2% accurate, 10.0× speedup

Math

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

FPCore

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

C

double code(double e, double v) {
	return e / (((e + 1.0) + (v * (v * ((e * -0.5) + (0.16666666666666666 - (e * -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)) + (0.16666666666666666d0 - (e * (-0.16666666666666666d0))))))) / v)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

   8. cos-lowering-cos.f64 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in v around 0

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

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

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

9. Simplified 51.8%

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

Alternative 12: 50.1% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around 0

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

   1. associate-/l* N/A

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

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

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

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

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

   4. +-commutative N/A

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

   5. +-lowering-+.f64 50.6%

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

5. Simplified 50.6%

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

Alternative 13: 49.6% 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. associate-/l* N/A

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

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

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

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

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

   4. +-commutative N/A

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

   5. +-lowering-+.f64 50.6%

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

5. Simplified 50.6%

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

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. sub-neg N/A

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

   7. mul-1-neg N/A

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

   8. remove-double-neg 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. *-lowering-*.f64 50.0%

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

8. Simplified 50.0%

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

Alternative 14: 49.0% 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. associate-/l* N/A

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

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

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

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

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

   4. +-commutative N/A

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

   5. +-lowering-+.f64 50.6%

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

5. Simplified 50.6%

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

6. Taylor expanded in e around 0

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

   1. *-lowering-*.f64 49.9%

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

8. Simplified 49.9%

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

Alternative 15: 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. associate-/l* N/A

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

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

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

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

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

   4. +-commutative N/A

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

   5. +-lowering-+.f64 50.6%

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

5. Simplified 50.6%

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

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 2024138 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))