Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 6.5s

Alternatives: 8

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 2: 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. Step-by-step derivation

   1. associate-/l* 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 3: 98.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in v around 0 99.3%

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

   1. +-commutative 99.3%

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

4. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 4: 97.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-/l* 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in e around 0 97.9%

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

5. Final simplification 97.9%

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

Alternative 5: 52.5% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.7%

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

4. Taylor expanded in v around 0 55.4%

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

5. Final simplification 55.4%

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

Alternative 6: 51.4% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(v * 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. Taylor expanded in v around 0 99.3%

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

   1. +-commutative 99.3%

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

4. Simplified 99.3%

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

5. Taylor expanded in v around 0 54.3%

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

   1. associate-/l* 54.1%

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

   2. *-rgt-identity 54.1%

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

   3. associate-*r/ 54.2%

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

   4. +-commutative 54.2%

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

   5. associate-/r/ 54.3%

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

   6. *-commutative 54.3%

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

   7. associate-*r/ 54.3%

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

   8. *-rgt-identity 54.3%

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

   9. *-lft-identity 54.3%

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

   10. fma-def 54.3%

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

   11. metadata-eval 54.3%

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

   12. fma-neg 54.3%

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

   13. *-lft-identity 54.3%

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

7. Simplified 54.3%

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

8. Final simplification 54.3%

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

Alternative 7: 50.3% 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* 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in v around 0 54.2%

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

   1. +-commutative 54.2%

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

6. Simplified 54.2%

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

7. Taylor expanded in e around 0 52.9%

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

8. Final simplification 52.9%

   \[\leadsto e \cdot v \]

Alternative 8: 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* 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in v around 0 54.2%

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

   1. +-commutative 54.2%

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

6. Simplified 54.2%

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

7. Taylor expanded in e around inf 4.6%

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

8. Final simplification 4.6%

   \[\leadsto v \]

Reproduce

herbie shell --seed 2023178 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))