Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 10.8s

Alternatives: 17

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, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\sin v}{1 - \left(e \cdot e\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(v \cdot 2\right)\right)} \cdot \left(e \cdot \left(1 - e \cdot \cos v\right)\right) \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (*
  (/ (sin v) (- 1.0 (* (* e e) (+ 0.5 (* 0.5 (cos (* v 2.0)))))))
  (* e (- 1.0 (* e (cos v))))))

C

double code(double e, double v) {
	return (sin(v) / (1.0 - ((e * e) * (0.5 + (0.5 * cos((v * 2.0))))))) * (e * (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 - ((e * e) * (0.5d0 + (0.5d0 * cos((v * 2.0d0))))))) * (e * (1.0d0 - (e * cos(v))))
end function

Java

public static double code(double e, double v) {
	return (Math.sin(v) / (1.0 - ((e * e) * (0.5 + (0.5 * Math.cos((v * 2.0))))))) * (e * (1.0 - (e * Math.cos(v))));
}

Python

def code(e, v):
	return (math.sin(v) / (1.0 - ((e * e) * (0.5 + (0.5 * math.cos((v * 2.0))))))) * (e * (1.0 - (e * math.cos(v))))

Julia

function code(e, v)
	return Float64(Float64(sin(v) / Float64(1.0 - Float64(Float64(e * e) * Float64(0.5 + Float64(0.5 * cos(Float64(v * 2.0))))))) * Float64(e * Float64(1.0 - Float64(e * cos(v)))))
end

MATLAB

function tmp = code(e, v)
	tmp = (sin(v) / (1.0 - ((e * e) * (0.5 + (0.5 * cos((v * 2.0))))))) * (e * (1.0 - (e * cos(v))));
end

Wolfram

code[e_, v_] := N[(N[(N[Sin[v], $MachinePrecision] / N[(1.0 - N[(N[(e * e), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(v * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(e * N[(1.0 - N[(e * N[Cos[v], $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. *-commutative N/A

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

   3. flip-+ N/A

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

   4. associate-/r/ N/A

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

   5. associate-*l* N/A

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

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

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

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\sin v}{1 - \left(e \cdot e\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot v\right)\right)} \cdot \left(\left(1 - e \cdot \cos v\right) \cdot e\right)} \]

5. Final simplification 99.8%

   \[\leadsto \frac{\sin v}{1 - \left(e \cdot e\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(v \cdot 2\right)\right)} \cdot \left(e \cdot \left(1 - e \cdot \cos v\right)\right) \]
6. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(N[Sin[v], $MachinePrecision] * e), $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. Final simplification 99.8%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. 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 99.7%

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

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{e}{\frac{1 + e \cdot \cos v}{\sin v}}} \]
5. 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. 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 5: 74.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\frac{v \cdot e}{1 + e}\\ \mathbf{else}:\\ \;\;\;\;\sin v \cdot e\\ \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (if (<= v 5e-85) (/ (* v e) (+ 1.0 e)) (* (sin v) e)))

C

double code(double e, double v) {
	double tmp;
	if (v <= 5e-85) {
		tmp = (v * e) / (1.0 + e);
	} else {
		tmp = sin(v) * e;
	}
	return tmp;
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    real(8) :: tmp
    if (v <= 5d-85) then
        tmp = (v * e) / (1.0d0 + e)
    else
        tmp = sin(v) * e
    end if
    code = tmp
end function

Java

public static double code(double e, double v) {
	double tmp;
	if (v <= 5e-85) {
		tmp = (v * e) / (1.0 + e);
	} else {
		tmp = Math.sin(v) * e;
	}
	return tmp;
}

Python

def code(e, v):
	tmp = 0
	if v <= 5e-85:
		tmp = (v * e) / (1.0 + e)
	else:
		tmp = math.sin(v) * e
	return tmp

Julia

function code(e, v)
	tmp = 0.0
	if (v <= 5e-85)
		tmp = Float64(Float64(v * e) / Float64(1.0 + e));
	else
		tmp = Float64(sin(v) * e);
	end
	return tmp
end

MATLAB

function tmp_2 = code(e, v)
	tmp = 0.0;
	if (v <= 5e-85)
		tmp = (v * e) / (1.0 + e);
	else
		tmp = sin(v) * e;
	end
	tmp_2 = tmp;
end

Wolfram

code[e_, v_] := If[LessEqual[v, 5e-85], N[(N[(v * e), $MachinePrecision] / N[(1.0 + e), $MachinePrecision]), $MachinePrecision], N[(N[Sin[v], $MachinePrecision] * e), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 5.0000000000000002e-85

   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

      \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]
   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 68.5%

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

   5. Simplified 68.5%

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

   if 5.0000000000000002e-85 < v

   1. Initial program 99.7%

      \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
   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 98.6%

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

   5. Simplified 98.6%

      \[\leadsto \color{blue}{e \cdot \sin v} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\frac{v \cdot e}{1 + e}\\ \mathbf{else}:\\ \;\;\;\;\sin v \cdot e\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(N[Sin[v], $MachinePrecision] * e), $MachinePrecision] / N[(1.0 + e), $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. Final simplification 98.9%

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

Alternative 7: 98.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. 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. 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. +-commutative N/A

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

   7. +-lowering-+.f64 98.9%

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

7. Applied egg-rr 98.9%

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

8. Final simplification 98.9%

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

Alternative 8: 52.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

double code(double e, double v) {
	double t_0 = (0.16666666666666666 * (-1.0 - e)) - (e * -0.5);
	double t_1 = (e * -0.5) + ((1.0 + e) * 0.16666666666666666);
	return e / (((1.0 + e) + ((v * v) * (t_1 + ((v * v) * ((((e * 0.041666666666666664) + (-0.16666666666666666 * t_0)) + ((1.0 + e) * -0.008333333333333333)) + ((v * v) * ((e * -0.001388888888888889) + ((-0.16666666666666666 * (((-0.16666666666666666 * t_1) - (e * 0.041666666666666664)) + (-0.008333333333333333 * (-1.0 - e)))) + ((0.008333333333333333 * t_0) - (-0.0001984126984126984 + (e * -0.0001984126984126984))))))))))) / 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
    t_0 = (0.16666666666666666d0 * ((-1.0d0) - e)) - (e * (-0.5d0))
    t_1 = (e * (-0.5d0)) + ((1.0d0 + e) * 0.16666666666666666d0)
    code = e / (((1.0d0 + e) + ((v * v) * (t_1 + ((v * v) * ((((e * 0.041666666666666664d0) + ((-0.16666666666666666d0) * t_0)) + ((1.0d0 + e) * (-0.008333333333333333d0))) + ((v * v) * ((e * (-0.001388888888888889d0)) + (((-0.16666666666666666d0) * ((((-0.16666666666666666d0) * t_1) - (e * 0.041666666666666664d0)) + ((-0.008333333333333333d0) * ((-1.0d0) - e)))) + ((0.008333333333333333d0 * t_0) - ((-0.0001984126984126984d0) + (e * (-0.0001984126984126984d0)))))))))))) / v)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := Block[{t$95$0 = N[(N[(0.16666666666666666 * N[(-1.0 - e), $MachinePrecision]), $MachinePrecision] - N[(e * -0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(e * -0.5), $MachinePrecision] + N[(N[(1.0 + e), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, N[(e / N[(N[(N[(1.0 + e), $MachinePrecision] + N[(N[(v * v), $MachinePrecision] * N[(t$95$1 + N[(N[(v * v), $MachinePrecision] * N[(N[(N[(N[(e * 0.041666666666666664), $MachinePrecision] + N[(-0.16666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + e), $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[(v * v), $MachinePrecision] * N[(N[(e * -0.001388888888888889), $MachinePrecision] + N[(N[(-0.16666666666666666 * N[(N[(N[(-0.16666666666666666 * t$95$1), $MachinePrecision] - N[(e * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + N[(-0.008333333333333333 * N[(-1.0 - e), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.008333333333333333 * t$95$0), $MachinePrecision] - N[(-0.0001984126984126984 + N[(e * -0.0001984126984126984), $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 99.7%

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

4. Applied egg-rr 99.7%

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

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 57.8%

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

8. Final simplification 57.8%

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

Alternative 9: 52.5% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e / N[(N[(N[(1.0 + e), $MachinePrecision] + N[(N[(v * v), $MachinePrecision] * N[(N[(e * -0.5), $MachinePrecision] + N[(N[(N[(1.0 + e), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + N[(N[(v * v), $MachinePrecision] * N[(N[(N[(e * 0.041666666666666664), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(0.16666666666666666 * N[(-1.0 - e), $MachinePrecision]), $MachinePrecision] - N[(e * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + e), $MachinePrecision] * -0.008333333333333333), $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 99.7%

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

4. Applied egg-rr 99.7%

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

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 57.7%

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

8. Final simplification 57.7%

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

Alternative 10: 52.7% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/l* N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

   9. sin-lowering-sin.f64 99.7%

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

4. Applied egg-rr 99.7%

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

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. +-commutative N/A

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

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

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

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

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

   7. unpow2 N/A

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

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

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

   9. sub-neg N/A

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

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

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

   11. *-commutative N/A

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

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

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

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

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

   14. metadata-eval N/A

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

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

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

   16. +-commutative N/A

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

   17. +-lowering-+.f64 57.5%

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

7. Simplified 57.5%

   \[\leadsto \frac{e}{\color{blue}{\frac{\left(e + 1\right) + \left(v \cdot v\right) \cdot \left(e \cdot -0.5 + 0.16666666666666666 \cdot \left(e + 1\right)\right)}{v}}} \]
8. Step-by-step derivation

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. clear-num N/A

      \[\leadsto \frac{v}{\left(e + 1\right) + \left(v \cdot v\right) \cdot \left(e \cdot \frac{-1}{2} + \frac{1}{6} \cdot \left(e + 1\right)\right)} \cdot e \]

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

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

9. Applied egg-rr 57.6%

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

10. Final simplification 57.6%

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

Alternative 11: 52.6% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e / N[(N[(N[(1.0 + e), $MachinePrecision] + N[(N[(v * v), $MachinePrecision] * N[(N[(e * -0.5), $MachinePrecision] + 0.16666666666666666), $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 99.7%

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

4. Applied egg-rr 99.7%

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

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. +-commutative N/A

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

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

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

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

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

   7. unpow2 N/A

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

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

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

   9. sub-neg N/A

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

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

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

   11. *-commutative N/A

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

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

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

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

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

   14. metadata-eval N/A

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

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

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

   16. +-commutative N/A

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

   17. +-lowering-+.f64 57.5%

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

7. Simplified 57.5%

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

8. Taylor expanded in e around 0

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

10. Simplified 57.5%

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

11. Final simplification 57.5%

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

Alternative 12: 52.2% accurate, 13.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e / N[(N[(N[(1.0 + e), $MachinePrecision] + N[(N[(v * v), $MachinePrecision] * N[(e * -0.3333333333333333), $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 99.7%

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

4. Applied egg-rr 99.7%

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

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. +-commutative N/A

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

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

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

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

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

   7. unpow2 N/A

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

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

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

   9. sub-neg N/A

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

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

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

   11. *-commutative N/A

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

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

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

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

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

   14. metadata-eval N/A

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

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

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

   16. +-commutative N/A

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

   17. +-lowering-+.f64 57.5%

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

7. Simplified 57.5%

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

8. Taylor expanded in e around inf

   \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(e, 1\right), \color{blue}{\left(\frac{-1}{3} \cdot \left(e \cdot {v}^{2}\right)\right)}\right), v\right)\right) \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(e, 1\right), \left(\left(\frac{-1}{3} \cdot e\right) \cdot {v}^{2}\right)\right), v\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(e, 1\right), \mathsf{*.f64}\left(\left(\frac{-1}{3} \cdot e\right), \left({v}^{2}\right)\right)\right), v\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(e, 1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, e\right), \left({v}^{2}\right)\right)\right), v\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(e, 1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, e\right), \left(v \cdot v\right)\right)\right), v\right)\right) \]

   5. *-lowering-*.f64 56.9%

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(e, 1\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, e\right), \mathsf{*.f64}\left(v, v\right)\right)\right), v\right)\right) \]

10. Simplified 56.9%

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

11. Final simplification 56.9%

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

Alternative 13: 51.5% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(v * e), $MachinePrecision] / N[(1.0 + e), $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 56.2%

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

5. Simplified 56.2%

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

6. Final simplification 56.2%

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

Alternative 14: 51.4% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e / N[(N[(1.0 + e), $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 99.7%

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in v around 0

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

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

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

   2. +-commutative N/A

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

   3. +-lowering-+.f64 56.1%

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

7. Simplified 56.1%

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

8. Final simplification 56.1%

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

Alternative 15: 51.0% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around 0

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

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

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

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

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

   3. +-commutative N/A

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

   4. +-lowering-+.f64 56.2%

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

5. Simplified 56.2%

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

6. Taylor expanded in e around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

   4. sub-neg 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. associate-*r* N/A

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

   11. *-lft-identity N/A

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

   12. distribute-rgt-in N/A

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

   13. neg-mul-1 N/A

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

   14. sub-neg N/A

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

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

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

   16. --lowering--.f64 55.1%

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

8. Simplified 55.1%

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

Alternative 16: 50.6% accurate, 69.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(v * e), $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 56.2%

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

5. Simplified 56.2%

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

6. Taylor expanded in e around 0

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

   1. *-lowering-*.f64 54.5%

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

8. Simplified 54.5%

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

9. Final simplification 54.5%

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

Alternative 17: 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 56.2%

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

5. Simplified 56.2%

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

6. Taylor expanded in e around inf

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

8. Simplified 4.8%

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

Reproduce

herbie shell --seed 2024158 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))