Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 9.6s

Alternatives: 8

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

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

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. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. +-commutative N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. cos-lowering-cos.f64 N/A

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

   9. sin-lowering-sin.f64 99.8

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

4. Applied egg-rr 99.8%

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

Alternative 2: 98.7% accurate, 1.9× 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

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

   1. +-lowering-+.f64 99.0

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

5. Simplified 99.0%

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

6. Final simplification 99.0%

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

Alternative 3: 98.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] * N[(1.0 - 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 e around 0

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

   1. distribute-lft-in N/A

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

   2. mul-1-neg N/A

      \[\leadsto e \cdot \sin v + e \cdot \color{blue}{\left(\mathsf{neg}\left(e \cdot \left(\cos v \cdot \sin v\right)\right)\right)} \]

   3. distribute-rgt-neg-out N/A

      \[\leadsto e \cdot \sin v + \color{blue}{\left(\mathsf{neg}\left(e \cdot \left(e \cdot \left(\cos v \cdot \sin v\right)\right)\right)\right)} \]

   4. associate-*r* N/A

      \[\leadsto e \cdot \sin v + \left(\mathsf{neg}\left(e \cdot \color{blue}{\left(\left(e \cdot \cos v\right) \cdot \sin v\right)}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto e \cdot \sin v + \left(\mathsf{neg}\left(e \cdot \left(\color{blue}{\left(\cos v \cdot e\right)} \cdot \sin v\right)\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto e \cdot \sin v + \left(\mathsf{neg}\left(e \cdot \color{blue}{\left(\cos v \cdot \left(e \cdot \sin v\right)\right)}\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto e \cdot \sin v + \left(\mathsf{neg}\left(\color{blue}{\left(e \cdot \cos v\right) \cdot \left(e \cdot \sin v\right)}\right)\right) \]

   8. distribute-lft-neg-in N/A

      \[\leadsto e \cdot \sin v + \color{blue}{\left(\mathsf{neg}\left(e \cdot \cos v\right)\right) \cdot \left(e \cdot \sin v\right)} \]

   9. distribute-rgt1-in N/A

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

   10. *-lowering-*.f64 N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. mul-1-neg N/A

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

   13. accelerator-lowering-fma.f64 N/A

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

   14. mul-1-neg N/A

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

   15. neg-sub0 N/A

       \[\leadsto \mathsf{fma}\left(e, \color{blue}{0 - \cos v}, 1\right) \cdot \left(e \cdot \sin v\right) \]

   16. --lowering--.f64 N/A

       \[\leadsto \mathsf{fma}\left(e, \color{blue}{0 - \cos v}, 1\right) \cdot \left(e \cdot \sin v\right) \]

   17. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{fma}\left(e, 0 - \color{blue}{\cos v}, 1\right) \cdot \left(e \cdot \sin v\right) \]

   18. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(e, 0 - \cos v, 1\right) \cdot \color{blue}{\left(e \cdot \sin v\right)} \]

5. Simplified 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(e, 0 - \cos v, 1\right) \cdot \left(e \cdot \sin v\right)} \]

6. Taylor expanded in v around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. --lowering--.f64 98.5

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

8. Simplified 98.5%

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

9. Final simplification 98.5%

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

Alternative 4: 97.8% accurate, 2.1× 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

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

   1. *-lowering-*.f64 N/A

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

   2. sin-lowering-sin.f64 98.1

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

5. Simplified 98.1%

   \[\leadsto \color{blue}{e \cdot \sin v} \]
6. Add Preprocessing

Alternative 5: 51.6% accurate, 11.3× 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

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

   1. /-lowering-/.f64 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. +-lowering-+.f64 56.5

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

5. Simplified 56.5%

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

   1. clear-num N/A

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

   2. associate-/r* N/A

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

   3. associate-/r/ N/A

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

   4. clear-num N/A

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

   5. *-lowering-*.f64 N/A

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

   6. /-lowering-/.f64 N/A

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

   7. +-commutative N/A

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

   8. +-lowering-+.f64 56.5

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

7. Applied egg-rr 56.5%

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

8. Final simplification 56.5%

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

Alternative 6: 51.2% accurate, 16.1× 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

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

   1. /-lowering-/.f64 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. +-lowering-+.f64 56.5

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

5. Simplified 56.5%

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

6. Taylor expanded in e around 0

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

   1. *-lowering-*.f64 N/A

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

   2. *-lft-identity N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-in N/A

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

   5. *-lowering-*.f64 N/A

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

   6. mul-1-neg N/A

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

   7. unsub-neg N/A

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

   8. --lowering--.f64 56.1

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

8. Simplified 56.1%

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

Alternative 7: 50.7% accurate, 37.5× 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 e around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sin-lowering-sin.f64 98.1

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

5. Simplified 98.1%

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

6. Taylor expanded in v around 0

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

8. Simplified 55.6%

   \[\leadsto e \cdot \color{blue}{v} \]
9. Add Preprocessing

Alternative 8: 4.5% accurate, 225.0× 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

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

   1. /-lowering-/.f64 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. +-lowering-+.f64 56.5

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

5. Simplified 56.5%

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

6. Taylor expanded in e around inf

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

8. Simplified 4.7%

   \[\leadsto \color{blue}{v} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024197 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))