Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 8.3s

Alternatives: 12

Speedup: N/A×

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. clear-num 99.1%

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

   2. associate-/r/ 99.6%

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

   3. clear-num 99.8%

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

   4. +-commutative 99.8%

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

   5. fma-udef 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in v around inf 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. 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 inf 99.8%

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

   1. associate-/l* 99.7%

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

   2. +-commutative 99.7%

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

   3. *-commutative 99.7%

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

   4. fma-udef 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in e around inf 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 98.7% accurate, 2.0× speedup

Math

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

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(sin(v) * Float64(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[Sin[v], $MachinePrecision] * N[(e / 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. Taylor expanded in v around 0 98.9%

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

   1. +-commutative 98.9%

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

6. Simplified 98.9%

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

7. Final simplification 98.9%

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

Alternative 4: 97.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[Sin[v], $MachinePrecision] * e), $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 98.1%

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

5. Final simplification 98.1%

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

Alternative 5: 51.9% 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 50.1%

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

5. Final simplification 50.1%

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

Alternative 6: 51.0% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(1.0 / N[(N[(1.0 / v), $MachinePrecision] + N[(N[(1.0 / N[(v * e), $MachinePrecision]), $MachinePrecision] - N[(v * N[(0.5 + N[(-0.16666666666666666 / e), $MachinePrecision]), $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/ 99.8%

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

   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. fma-udef 99.8%

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

   2. +-commutative 99.8%

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

   3. associate-/r/ 99.7%

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

   4. clear-num 99.1%

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

   5. +-commutative 99.1%

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

   6. fma-udef 99.1%

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

5. Applied egg-rr 99.1%

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

6. Taylor expanded in v around 0 49.8%

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

   1. mul-1-neg 49.8%

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

   2. *-commutative 49.8%

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

   3. +-commutative 49.8%

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

8. Simplified 49.8%

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

9. Taylor expanded in e around 0 49.8%

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

10. Final simplification 49.8%

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

Alternative 7: 51.0% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(1.0 / N[(N[(1.0 / v), $MachinePrecision] + N[(N[(1.0 / N[(v * e), $MachinePrecision]), $MachinePrecision] - N[(-0.16666666666666666 * N[(v / e), $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/ 99.8%

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

   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. fma-udef 99.8%

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

   2. +-commutative 99.8%

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

   3. associate-/r/ 99.7%

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

   4. clear-num 99.1%

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

   5. +-commutative 99.1%

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

   6. fma-udef 99.1%

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

5. Applied egg-rr 99.1%

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

6. Taylor expanded in v around 0 49.8%

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

   1. mul-1-neg 49.8%

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

   2. *-commutative 49.8%

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

   3. +-commutative 49.8%

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

8. Simplified 49.8%

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

9. Taylor expanded in e around 0 49.8%

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

10. Final simplification 49.8%

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

Alternative 8: 50.9% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \frac{e}{\frac{1}{v} - v \cdot -0.16666666666666666} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e / N[(N[(1.0 / v), $MachinePrecision] - N[(v * -0.16666666666666666), $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. fma-udef 99.8%

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

   2. +-commutative 99.8%

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

   3. associate-/r/ 99.7%

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

   4. clear-num 99.1%

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

   5. +-commutative 99.1%

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

   6. fma-udef 99.1%

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

5. Applied egg-rr 99.1%

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

6. Taylor expanded in v around 0 49.8%

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

   1. mul-1-neg 49.8%

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

   2. *-commutative 49.8%

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

   3. +-commutative 49.8%

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

8. Simplified 49.8%

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

9. Taylor expanded in e around 0 49.3%

   \[\leadsto \color{blue}{\frac{e}{\frac{1}{v} - -0.16666666666666666 \cdot v}} \]

10. Final simplification 49.3%

    \[\leadsto \frac{e}{\frac{1}{v} - v \cdot -0.16666666666666666} \]

Alternative 9: 50.2% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(v * N[(e - N[(e * 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* 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in v around 0 48.9%

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

   1. +-commutative 48.9%

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

6. Simplified 48.9%

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

7. Taylor expanded in e around 0 48.4%

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

   1. mul-1-neg 48.4%

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

   2. unsub-neg 48.4%

      \[\leadsto \color{blue}{v \cdot e - v \cdot {e}^{2}} \]

   3. unpow2 48.4%

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

9. Simplified 48.4%

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

10. Taylor expanded in v around 0 48.4%

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

    1. unpow2 48.4%

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

12. Simplified 48.4%

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

13. Final simplification 48.4%

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

Alternative 10: 50.8% accurate, 29.9× speedup

Math

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

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(v * Float64(e / Float64(1.0 + e)))
end

MATLAB

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

Wolfram

code[e_, v_] := N[(v * N[(e / 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* 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in v around 0 48.9%

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

   1. +-commutative 48.9%

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

6. Simplified 48.9%

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

   1. associate-/r/ 49.0%

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

8. Applied egg-rr 49.0%

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

9. Final simplification 49.0%

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

Alternative 11: 49.8% 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. 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 48.9%

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

   1. +-commutative 48.9%

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

6. Simplified 48.9%

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

7. Taylor expanded in e around 0 48.1%

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

8. Final simplification 48.1%

   \[\leadsto v \cdot e \]

Alternative 12: 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 48.9%

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

   1. +-commutative 48.9%

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

6. Simplified 48.9%

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

7. Taylor expanded in e around inf 4.4%

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

8. Final simplification 4.4%

   \[\leadsto v \]

Reproduce

herbie shell --seed 2023182 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))