Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 9.4s

Alternatives: 11

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

   6. +-commutative N/A

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

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

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

   8. cos-lowering-cos.f64 99.8

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (e v) :precision binary64 (* (sin v) (fma (cos v) (* e (- e)) e)))

C

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

Julia

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

Wolfram

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

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. mul-1-neg N/A

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

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

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

   7. unpow2 N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{{e}^{2}}\right)\right) \cdot \left(\cos v \cdot \sin v\right) + e \cdot \sin v \]

   8. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left({e}^{2}\right)\right) \cdot \cos v\right) \cdot \sin v} + e \cdot \sin v \]

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

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left({e}^{2} \cdot \cos v\right)\right)} \cdot \sin v + e \cdot \sin v \]

   10. distribute-rgt-out N/A

       \[\leadsto \color{blue}{\sin v \cdot \left(\left(\mathsf{neg}\left({e}^{2} \cdot \cos v\right)\right) + e\right)} \]

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

       \[\leadsto \color{blue}{\sin v \cdot \left(\left(\mathsf{neg}\left({e}^{2} \cdot \cos v\right)\right) + e\right)} \]

   12. sin-lowering-sin.f64 N/A

       \[\leadsto \color{blue}{\sin v} \cdot \left(\left(\mathsf{neg}\left({e}^{2} \cdot \cos v\right)\right) + e\right) \]

   13. *-commutative N/A

       \[\leadsto \sin v \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\cos v \cdot {e}^{2}}\right)\right) + e\right) \]

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

       \[\leadsto \sin v \cdot \left(\color{blue}{\cos v \cdot \left(\mathsf{neg}\left({e}^{2}\right)\right)} + e\right) \]

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

       \[\leadsto \sin v \cdot \color{blue}{\mathsf{fma}\left(\cos v, \mathsf{neg}\left({e}^{2}\right), e\right)} \]

5. Simplified 99.0%

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

Alternative 3: 97.8% accurate, 2.1× 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. 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 98.2

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

5. Simplified 98.2%

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

6. Final simplification 98.2%

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

Alternative 4: 52.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\mathsf{fma}\left(v \cdot v, \mathsf{fma}\left(v \cdot v, 0.041666666666666664 + \left(-0.019444444444444445 \cdot \frac{-1 - e}{e} - 0.08333333333333333\right), -0.5 - \frac{\mathsf{fma}\left(e, -0.16666666666666666, -0.16666666666666666\right)}{e}\right), 1 + \frac{1}{e}\right)}{v}} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (/
  1.0
  (/
   (fma
    (* v v)
    (fma
     (* v v)
     (+
      0.041666666666666664
      (- (* -0.019444444444444445 (/ (- -1.0 e) e)) 0.08333333333333333))
     (- -0.5 (/ (fma e -0.16666666666666666 -0.16666666666666666) e)))
    (+ 1.0 (/ 1.0 e)))
   v)))

C

double code(double e, double v) {
	return 1.0 / (fma((v * v), fma((v * v), (0.041666666666666664 + ((-0.019444444444444445 * ((-1.0 - e) / e)) - 0.08333333333333333)), (-0.5 - (fma(e, -0.16666666666666666, -0.16666666666666666) / e))), (1.0 + (1.0 / e))) / v);
}

Julia

function code(e, v)
	return Float64(1.0 / Float64(fma(Float64(v * v), fma(Float64(v * v), Float64(0.041666666666666664 + Float64(Float64(-0.019444444444444445 * Float64(Float64(-1.0 - e) / e)) - 0.08333333333333333)), Float64(-0.5 - Float64(fma(e, -0.16666666666666666, -0.16666666666666666) / e))), Float64(1.0 + Float64(1.0 / e))) / v))
end

Wolfram

code[e_, v_] := N[(1.0 / N[(N[(N[(v * v), $MachinePrecision] * N[(N[(v * v), $MachinePrecision] * N[(0.041666666666666664 + N[(N[(-0.019444444444444445 * N[(N[(-1.0 - e), $MachinePrecision] / e), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + N[(-0.5 - N[(N[(e * -0.16666666666666666 + -0.16666666666666666), $MachinePrecision] / e), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(1.0 / e), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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. clear-num N/A

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

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

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

   3. /-lowering-/.f64 N/A

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

   4. +-commutative N/A

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

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

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

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

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

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

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

   8. sin-lowering-sin.f64 99.0

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

4. Applied egg-rr 99.0%

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

5. Taylor expanded in v around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{1 + \left({v}^{2} \cdot \left({v}^{2} \cdot \left(\frac{1}{24} - \left(\frac{1}{120} \cdot \frac{1 + e}{e} + \frac{1}{6} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \frac{1 + e}{e}\right)\right)\right) - \left(\frac{1}{2} + \frac{-1}{6} \cdot \frac{1 + e}{e}\right)\right) + \frac{1}{e}\right)}{v}}} \]
6. Step-by-step derivation

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

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + \left({v}^{2} \cdot \left({v}^{2} \cdot \left(\frac{1}{24} - \left(\frac{1}{120} \cdot \frac{1 + e}{e} + \frac{1}{6} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \frac{1 + e}{e}\right)\right)\right) - \left(\frac{1}{2} + \frac{-1}{6} \cdot \frac{1 + e}{e}\right)\right) + \frac{1}{e}\right)}{v}}} \]

7. Simplified 59.3%

   \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(v \cdot v, \mathsf{fma}\left(v \cdot v, 0.041666666666666664 - \left(0.08333333333333333 + \frac{e + 1}{e} \cdot -0.019444444444444445\right), -0.5 - \frac{\mathsf{fma}\left(e, -0.16666666666666666, -0.16666666666666666\right)}{e}\right), 1 + \frac{1}{e}\right)}{v}}} \]

8. Final simplification 59.3%

   \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(v \cdot v, \mathsf{fma}\left(v \cdot v, 0.041666666666666664 + \left(-0.019444444444444445 \cdot \frac{-1 - e}{e} - 0.08333333333333333\right), -0.5 - \frac{\mathsf{fma}\left(e, -0.16666666666666666, -0.16666666666666666\right)}{e}\right), 1 + \frac{1}{e}\right)}{v}} \]
9. Add Preprocessing

Alternative 5: 52.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 8.2 \cdot 10^{-10}:\\ \;\;\;\;v \cdot \frac{e}{e + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\frac{v \cdot v}{e}, \mathsf{fma}\left(v \cdot v, 0.019444444444444445, 0.16666666666666666\right), \frac{1}{e}\right)}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (if (<= v 8.2e-10)
   (* v (/ e (+ e 1.0)))
   (/
    1.0
    (/
     (fma
      (/ (* v v) e)
      (fma (* v v) 0.019444444444444445 0.16666666666666666)
      (/ 1.0 e))
     v))))

C

double code(double e, double v) {
	double tmp;
	if (v <= 8.2e-10) {
		tmp = v * (e / (e + 1.0));
	} else {
		tmp = 1.0 / (fma(((v * v) / e), fma((v * v), 0.019444444444444445, 0.16666666666666666), (1.0 / e)) / v);
	}
	return tmp;
}

Julia

function code(e, v)
	tmp = 0.0
	if (v <= 8.2e-10)
		tmp = Float64(v * Float64(e / Float64(e + 1.0)));
	else
		tmp = Float64(1.0 / Float64(fma(Float64(Float64(v * v) / e), fma(Float64(v * v), 0.019444444444444445, 0.16666666666666666), Float64(1.0 / e)) / v));
	end
	return tmp
end

Wolfram

code[e_, v_] := If[LessEqual[v, 8.2e-10], N[(v * N[(e / N[(e + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(v * v), $MachinePrecision] / e), $MachinePrecision] * N[(N[(v * v), $MachinePrecision] * 0.019444444444444445 + 0.16666666666666666), $MachinePrecision] + N[(1.0 / e), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 8.1999999999999996e-10

   1. Initial program 99.9%

      \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
   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 71.2

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

   5. Simplified 71.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      7. +-lowering-+.f64 71.2

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

   7. Applied egg-rr 71.2%

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

   if 8.1999999999999996e-10 < v

   1. Initial program 99.7%

      \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
   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 97.3

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

   5. Simplified 97.3%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. sin-lowering-sin.f64 97.2

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

   7. Applied egg-rr 97.2%

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

   8. Taylor expanded in v around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{{v}^{2} \cdot \left(\frac{7}{360} \cdot \frac{{v}^{2}}{e} + \frac{1}{6} \cdot \frac{1}{e}\right) + \frac{1}{e}}{v}}} \]
   9. Step-by-step derivation

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

         \[\leadsto \frac{1}{\color{blue}{\frac{{v}^{2} \cdot \left(\frac{7}{360} \cdot \frac{{v}^{2}}{e} + \frac{1}{6} \cdot \frac{1}{e}\right) + \frac{1}{e}}{v}}} \]

   10. Simplified 16.8%

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\frac{v \cdot v}{e}, \mathsf{fma}\left(v \cdot v, 0.019444444444444445, 0.16666666666666666\right), \frac{1}{e}\right)}{v}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 8.2 \cdot 10^{-10}:\\ \;\;\;\;v \cdot \frac{e}{e + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\frac{v \cdot v}{e}, \mathsf{fma}\left(v \cdot v, 0.019444444444444445, 0.16666666666666666\right), \frac{1}{e}\right)}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[e_, v_] := N[(1.0 / N[(N[(1.0 + N[(N[(v * v), $MachinePrecision] * N[(-0.5 - N[(N[(e * -0.16666666666666666 + -0.16666666666666666), $MachinePrecision] / e), $MachinePrecision]), $MachinePrecision] + N[(1.0 / e), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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. clear-num N/A

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

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

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

   3. /-lowering-/.f64 N/A

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

   4. +-commutative N/A

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

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

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

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

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

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

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

   8. sin-lowering-sin.f64 99.0

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

4. Applied egg-rr 99.0%

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

5. Taylor expanded in v around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{1 + \left(-1 \cdot \left({v}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \frac{1 + e}{e}\right)\right) + \frac{1}{e}\right)}{v}}} \]
6. Step-by-step derivation

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

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + \left(-1 \cdot \left({v}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot \frac{1 + e}{e}\right)\right) + \frac{1}{e}\right)}{v}}} \]

7. Simplified 59.0%

   \[\leadsto \frac{1}{\color{blue}{\frac{1 + \mathsf{fma}\left(v \cdot v, -0.5 - \frac{\mathsf{fma}\left(e, -0.16666666666666666, -0.16666666666666666\right)}{e}, \frac{1}{e}\right)}{v}}} \]
8. Add Preprocessing

Alternative 7: 52.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 8.2 \cdot 10^{-10}:\\ \;\;\;\;v \cdot \frac{e}{e + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{e} \cdot \mathsf{fma}\left(v \cdot v, 0.16666666666666666, 1\right)}{v}}\\ \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (if (<= v 8.2e-10)
   (* v (/ e (+ e 1.0)))
   (/ 1.0 (/ (* (/ 1.0 e) (fma (* v v) 0.16666666666666666 1.0)) v))))

C

double code(double e, double v) {
	double tmp;
	if (v <= 8.2e-10) {
		tmp = v * (e / (e + 1.0));
	} else {
		tmp = 1.0 / (((1.0 / e) * fma((v * v), 0.16666666666666666, 1.0)) / v);
	}
	return tmp;
}

Julia

function code(e, v)
	tmp = 0.0
	if (v <= 8.2e-10)
		tmp = Float64(v * Float64(e / Float64(e + 1.0)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(1.0 / e) * fma(Float64(v * v), 0.16666666666666666, 1.0)) / v));
	end
	return tmp
end

Wolfram

code[e_, v_] := If[LessEqual[v, 8.2e-10], N[(v * N[(e / N[(e + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(1.0 / e), $MachinePrecision] * N[(N[(v * v), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 8.1999999999999996e-10

   1. Initial program 99.9%

      \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
   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 71.2

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

   5. Simplified 71.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      7. +-lowering-+.f64 71.2

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

   7. Applied egg-rr 71.2%

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

   if 8.1999999999999996e-10 < v

   1. Initial program 99.7%

      \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
   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 97.3

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

   5. Simplified 97.3%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. sin-lowering-sin.f64 97.2

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

   7. Applied egg-rr 97.2%

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

   8. Taylor expanded in v around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{1}{6} \cdot \frac{{v}^{2}}{e} + \frac{1}{e}}{v}}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{\frac{1}{6} \cdot {v}^{2}}{e}} + \frac{1}{e}}{v}} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\frac{\frac{1}{6}}{e} \cdot {v}^{2}} + \frac{1}{e}}{v}} \]

      3. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\frac{\color{blue}{\frac{1}{6} \cdot 1}}{e} \cdot {v}^{2} + \frac{1}{e}}{v}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(\frac{1}{6} \cdot \frac{1}{e}\right)} \cdot {v}^{2} + \frac{1}{e}}{v}} \]

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

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(\frac{1}{6} \cdot \frac{1}{e}\right) \cdot {v}^{2} + \frac{1}{e}}{v}}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{{v}^{2} \cdot \left(\frac{1}{6} \cdot \frac{1}{e}\right)} + \frac{1}{e}}{v}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left({v}^{2} \cdot \frac{1}{6}\right) \cdot \frac{1}{e}} + \frac{1}{e}}{v}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(\frac{1}{6} \cdot {v}^{2}\right)} \cdot \frac{1}{e} + \frac{1}{e}}{v}} \]

      9. distribute-lft1-in N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(\frac{1}{6} \cdot {v}^{2} + 1\right) \cdot \frac{1}{e}}}{v}} \]

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

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(\frac{1}{6} \cdot {v}^{2} + 1\right) \cdot \frac{1}{e}}}{v}} \]

      11. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\left(\color{blue}{{v}^{2} \cdot \frac{1}{6}} + 1\right) \cdot \frac{1}{e}}{v}} \]

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

          \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left({v}^{2}, \frac{1}{6}, 1\right)} \cdot \frac{1}{e}}{v}} \]

      13. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{v \cdot v}, \frac{1}{6}, 1\right) \cdot \frac{1}{e}}{v}} \]

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

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{v \cdot v}, \frac{1}{6}, 1\right) \cdot \frac{1}{e}}{v}} \]

      15. /-lowering-/.f64 16.5

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(v \cdot v, 0.16666666666666666, 1\right) \cdot \color{blue}{\frac{1}{e}}}{v}} \]

   10. Simplified 16.5%

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(v \cdot v, 0.16666666666666666, 1\right) \cdot \frac{1}{e}}{v}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 8.2 \cdot 10^{-10}:\\ \;\;\;\;v \cdot \frac{e}{e + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{e} \cdot \mathsf{fma}\left(v \cdot v, 0.16666666666666666, 1\right)}{v}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.0% 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 58.1

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

5. Simplified 58.1%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

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

   7. +-lowering-+.f64 58.2

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

7. Applied egg-rr 58.2%

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

8. Final simplification 58.2%

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

Alternative 9: 51.5% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e * N[(v - N[(v * 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 58.1

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

5. Simplified 58.1%

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

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. mul-1-neg N/A

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

   3. unsub-neg N/A

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

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

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

   5. *-lowering-*.f64 57.8

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

8. Simplified 57.8%

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

9. Final simplification 57.8%

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

Alternative 10: 51.1% accurate, 37.5× 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. 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 98.2

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

5. Simplified 98.2%

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

6. Taylor expanded in v around 0

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

8. Simplified 57.3%

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

9. Final simplification 57.3%

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

Alternative 11: 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 58.1

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

5. Simplified 58.1%

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

6. Taylor expanded in e around inf

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

8. Simplified 5.0%

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

Reproduce

herbie shell --seed 2024199 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))