Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 7.3s

Alternatives: 11

Speedup: 1.0×

Specification

\[0 \leq e \land e \leq 1\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 3: 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. Step-by-step derivation

   1. associate-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{\frac{e}{\mathsf{fma}\left(e, \cos v, 1\right)} \cdot \sin v} \]
4. Step-by-step derivation

   1. *-commutative 99.8%

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

   2. clear-num 99.7%

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

   3. un-div-inv 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in e around inf 99.7%

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

7. Final simplification 99.7%

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

Alternative 4: 98.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in v around 0 99.3%

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

   1. +-commutative 99.3%

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

6. Simplified 99.3%

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

7. Final simplification 99.3%

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

Alternative 5: 97.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 99.8%

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

   2. +-commutative 99.8%

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

   3. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in e around 0 98.4%

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

5. Final simplification 98.4%

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

Alternative 6: 52.3% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/l* 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in v around 0 57.2%

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

5. Final simplification 57.2%

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

Alternative 7: 52.3% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/l* 99.6%

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

3. Simplified 99.6%

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

   1. frac-2neg 99.6%

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

   2. div-inv 99.7%

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

   3. distribute-neg-frac 99.7%

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

   4. +-commutative 99.7%

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

   5. fma-udef 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in v around 0 57.3%

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

   1. +-commutative 57.3%

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

   2. +-commutative 57.3%

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

   3. distribute-lft-out 57.3%

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

   4. *-lft-identity 57.3%

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

   5. associate-*r/ 57.3%

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

   6. associate-*l/ 57.3%

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

   7. distribute-rgt-in 57.3%

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

   8. associate-*r/ 57.3%

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

   9. *-rgt-identity 57.3%

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

   10. *-lft-identity 57.3%

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

   11. +-commutative 57.3%

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

   12. *-commutative 57.3%

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

   13. *-commutative 57.3%

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

   14. distribute-rgt-in 57.3%

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

   15. metadata-eval 57.3%

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

8. Simplified 57.3%

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

9. Final simplification 57.3%

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

Alternative 8: 51.9% accurate, 13.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/l* 99.6%

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

3. Simplified 99.6%

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

   1. frac-2neg 99.6%

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

   2. div-inv 99.7%

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

   3. distribute-neg-frac 99.7%

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

   4. +-commutative 99.7%

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

   5. fma-udef 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in v around 0 57.3%

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

   1. +-commutative 57.3%

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

   2. +-commutative 57.3%

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

   3. distribute-lft-out 57.3%

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

   4. *-lft-identity 57.3%

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

   5. associate-*r/ 57.3%

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

   6. associate-*l/ 57.3%

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

   7. distribute-rgt-in 57.3%

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

   8. associate-*r/ 57.3%

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

   9. *-rgt-identity 57.3%

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

   10. *-lft-identity 57.3%

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

   11. +-commutative 57.3%

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

   12. *-commutative 57.3%

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

   13. *-commutative 57.3%

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

   14. distribute-rgt-in 57.3%

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

   15. metadata-eval 57.3%

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

8. Simplified 57.3%

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

9. Taylor expanded in e around inf 56.9%

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

    1. *-commutative 56.9%

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

    2. associate-*l* 56.9%

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

11. Simplified 56.9%

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

12. Final simplification 56.9%

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

Alternative 9: 51.2% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/l* 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in v around 0 56.2%

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

   1. +-commutative 56.2%

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

6. Simplified 56.2%

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

   1. clear-num 54.5%

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

   2. associate-/r/ 56.3%

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

   3. clear-num 56.3%

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

8. Applied egg-rr 56.3%

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

9. Final simplification 56.3%

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

Alternative 10: 50.1% accurate, 69.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/l* 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in v around 0 56.2%

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

   1. +-commutative 56.2%

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

6. Simplified 56.2%

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

7. Taylor expanded in e around 0 55.5%

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

8. Final simplification 55.5%

   \[\leadsto e \cdot v \]

Alternative 11: 4.5% accurate, 209.0× speedup

Math

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

FPCore

(FPCore (e v) :precision binary64 v)

C

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

Fortran

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

Java

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

Python

def code(e, v):
	return v

Julia

function code(e, v)
	return v
end

MATLAB

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

Wolfram

code[e_, v_] := v

Derivation

1. Initial program 99.8%

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

   1. associate-/l* 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in v around 0 56.2%

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

   1. +-commutative 56.2%

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

6. Simplified 56.2%

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

7. Taylor expanded in e around inf 4.8%

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

8. Final simplification 4.8%

   \[\leadsto v \]

Reproduce

herbie shell --seed 2023199 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))