Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 7.2s

Alternatives: 9

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. Add Preprocessing

3. Final simplification 99.8%

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

Alternative 2: 98.5% accurate, 2.0× speedup

Math

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

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(Float64(e + 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[(N[(e + 1.0), $MachinePrecision] / 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 98.8%

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

   1. add-cube-cbrt 97.3%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{e \cdot \sin v}{1 + e}} \cdot \sqrt[3]{\frac{e \cdot \sin v}{1 + e}}\right) \cdot \sqrt[3]{\frac{e \cdot \sin v}{1 + e}}} \]

   2. pow3 97.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{e \cdot \sin v}{1 + e}}\right)}^{3}} \]

   3. *-commutative 97.3%

      \[\leadsto {\left(\sqrt[3]{\frac{\color{blue}{\sin v \cdot e}}{1 + e}}\right)}^{3} \]

   4. associate-/l* 97.3%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\sin v \cdot \frac{e}{1 + e}}}\right)}^{3} \]

   5. +-commutative 97.3%

      \[\leadsto {\left(\sqrt[3]{\sin v \cdot \frac{e}{\color{blue}{e + 1}}}\right)}^{3} \]

5. Applied egg-rr 97.3%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\sin v \cdot \frac{e}{e + 1}}\right)}^{3}} \]
6. Step-by-step derivation

   1. rem-cube-cbrt 98.8%

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

   2. clear-num 98.7%

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

   3. un-div-inv 98.6%

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

   4. +-commutative 98.6%

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

7. Applied egg-rr 98.6%

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

8. Final simplification 98.6%

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

Alternative 3: 98.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around 0 98.8%

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

4. Final simplification 98.8%

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

Alternative 4: 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. Add Preprocessing

3. Taylor expanded in e around 0 97.6%

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

4. Final simplification 97.6%

   \[\leadsto e \cdot \sin v \]
5. Add Preprocessing

Alternative 5: 50.9% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around 0 49.4%

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

   1. *-commutative 49.4%

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

   2. associate-*r/ 49.4%

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

   3. +-commutative 49.4%

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

5. Simplified 49.4%

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

   1. clear-num 49.3%

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

   2. +-commutative 49.3%

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

   3. associate-/r/ 49.4%

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

   4. +-commutative 49.4%

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

7. Applied egg-rr 49.4%

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

8. Final simplification 49.4%

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

Alternative 6: 50.4% 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 49.4%

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

   1. *-commutative 49.4%

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

   2. associate-*r/ 49.4%

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

   3. +-commutative 49.4%

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

5. Simplified 49.4%

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

6. Taylor expanded in e around 0 48.8%

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

   1. +-commutative 48.8%

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

   2. mul-1-neg 48.8%

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

   3. distribute-lft-neg-out 48.8%

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

   4. *-lft-identity 48.8%

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

   5. distribute-rgt-out 48.8%

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

   6. +-commutative 48.8%

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

   7. sub-neg 48.8%

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

8. Simplified 48.8%

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

9. Final simplification 48.8%

   \[\leadsto e \cdot \left(v \cdot \left(1 - e\right)\right) \]
10. Add Preprocessing

Alternative 7: 50.9% 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. Add Preprocessing

3. Taylor expanded in v around 0 49.4%

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

   1. *-commutative 49.4%

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

   2. associate-*r/ 49.4%

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

   3. +-commutative 49.4%

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

5. Simplified 49.4%

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

6. Final simplification 49.4%

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

Alternative 8: 49.9% accurate, 69.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around 0 49.4%

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

   1. *-commutative 49.4%

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

   2. associate-*r/ 49.4%

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

   3. +-commutative 49.4%

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

5. Simplified 49.4%

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

6. Taylor expanded in e around 0 48.2%

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

7. Final simplification 48.2%

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

Alternative 9: 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 49.4%

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

   1. *-commutative 49.4%

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

   2. associate-*r/ 49.4%

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

   3. +-commutative 49.4%

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

5. Simplified 49.4%

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

6. Taylor expanded in e around inf 4.5%

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

7. Final simplification 4.5%

   \[\leadsto v \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))