Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 7.1s

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, 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* 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 3: 98.6% 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. Taylor expanded in v around 0 99.1%

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

3. Taylor expanded in v around inf 99.1%

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

   1. associate-/l* 98.8%

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

   2. +-commutative 98.8%

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

5. Simplified 98.8%

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

6. Final simplification 98.8%

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

Alternative 4: 98.8% 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 99.1%

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

3. Final simplification 99.1%

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

Alternative 5: 97.9% 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* 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in e around 0 97.7%

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

5. Final simplification 97.7%

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

Alternative 6: 54.4% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e / N[(1.0 + N[(N[(e + 1.0), $MachinePrecision] / 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* 99.6%

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

3. Simplified 99.6%

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

   1. expm1-log1p-u 53.0%

      \[\leadsto \frac{e}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + e \cdot \cos v}{\sin v}\right)\right)}} \]

   2. +-commutative 53.0%

      \[\leadsto \frac{e}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{e \cdot \cos v + 1}}{\sin v}\right)\right)} \]

   3. fma-def 53.0%

      \[\leadsto \frac{e}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{fma}\left(e, \cos v, 1\right)}}{\sin v}\right)\right)} \]

5. Applied egg-rr 53.0%

   \[\leadsto \frac{e}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(e, \cos v, 1\right)}{\sin v}\right)\right)}} \]
6. Step-by-step derivation

   1. expm1-udef 52.9%

      \[\leadsto \frac{e}{\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(e, \cos v, 1\right)}{\sin v}\right)} - 1}} \]

   2. log1p-expm1-u 52.9%

      \[\leadsto \frac{e}{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(e, \cos v, 1\right)}{\sin v}\right)\right)\right)}} - 1} \]

   3. log1p-udef 52.9%

      \[\leadsto \frac{e}{e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(e, \cos v, 1\right)}{\sin v}\right)\right)\right)}} - 1} \]

   4. add-exp-log 52.9%

      \[\leadsto \frac{e}{\color{blue}{\left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(e, \cos v, 1\right)}{\sin v}\right)\right)\right)} - 1} \]

   5. expm1-log1p-u 99.5%

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

7. Applied egg-rr 99.5%

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

   1. associate--l+ 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in v around 0 56.2%

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

11. Final simplification 56.2%

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

Alternative 7: 50.2% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e * N[(v - N[(e * 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* 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in v around 0 52.7%

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

   1. associate-/l* 52.6%

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

   2. associate-/r/ 52.7%

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

   3. +-commutative 52.7%

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

6. Simplified 52.7%

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

7. Taylor expanded in e around 0 51.9%

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

   1. mul-1-neg 51.9%

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

   2. unsub-neg 51.9%

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

9. Simplified 51.9%

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

10. Final simplification 51.9%

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

Alternative 8: 50.6% 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 52.7%

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

   1. associate-/l* 52.6%

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

   2. associate-/r/ 52.7%

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

   3. +-commutative 52.7%

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

6. Simplified 52.7%

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

7. Final simplification 52.7%

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

Alternative 9: 50.6% accurate, 29.9× speedup

Math

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

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(e * v), $MachinePrecision] / N[(e + 1.0), $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. expm1-log1p-u 53.0%

      \[\leadsto \frac{e}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + e \cdot \cos v}{\sin v}\right)\right)}} \]

   2. +-commutative 53.0%

      \[\leadsto \frac{e}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{e \cdot \cos v + 1}}{\sin v}\right)\right)} \]

   3. fma-def 53.0%

      \[\leadsto \frac{e}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{fma}\left(e, \cos v, 1\right)}}{\sin v}\right)\right)} \]

5. Applied egg-rr 53.0%

   \[\leadsto \frac{e}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(e, \cos v, 1\right)}{\sin v}\right)\right)}} \]
6. Step-by-step derivation

   1. expm1-udef 52.9%

      \[\leadsto \frac{e}{\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(e, \cos v, 1\right)}{\sin v}\right)} - 1}} \]

   2. log1p-expm1-u 52.9%

      \[\leadsto \frac{e}{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(e, \cos v, 1\right)}{\sin v}\right)\right)\right)}} - 1} \]

   3. log1p-udef 52.9%

      \[\leadsto \frac{e}{e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(e, \cos v, 1\right)}{\sin v}\right)\right)\right)}} - 1} \]

   4. add-exp-log 52.9%

      \[\leadsto \frac{e}{\color{blue}{\left(1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(e, \cos v, 1\right)}{\sin v}\right)\right)\right)} - 1} \]

   5. expm1-log1p-u 99.5%

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

7. Applied egg-rr 99.5%

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

   1. associate--l+ 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in v around 0 52.7%

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

11. Final simplification 52.7%

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

Alternative 10: 49.7% 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 52.7%

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

   1. associate-/l* 52.6%

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

   2. associate-/r/ 52.7%

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

   3. +-commutative 52.7%

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

6. Simplified 52.7%

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

7. Taylor expanded in e around 0 51.3%

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

8. Final simplification 51.3%

   \[\leadsto e \cdot v \]

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 54.0%

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

5. Taylor expanded in e around inf 5.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{v} + -0.3333333333333333 \cdot v}} \]
6. Step-by-step derivation

   1. *-commutative 5.7%

      \[\leadsto \frac{1}{\frac{1}{v} + \color{blue}{v \cdot -0.3333333333333333}} \]

7. Simplified 5.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{v} + v \cdot -0.3333333333333333}} \]

8. Taylor expanded in v around 0 4.3%

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

9. Final simplification 4.3%

   \[\leadsto v \]

Reproduce

herbie shell --seed 2023240 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))