Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 9.8s

Alternatives: 12

Speedup: N/A×

Specification

\[0 \leq e \land e \leq 1\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

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. lift-sin.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 99.8

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 98.8% 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. lower-+.f64 98.4

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

5. Applied rewrites 98.4%

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

6. Final simplification 98.4%

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

Alternative 3: 98.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. lift-sin.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 99.8

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 98.4

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

7. Applied rewrites 98.4%

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

8. Final simplification 98.4%

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

Alternative 4: 98.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. 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. lower-*.f64 N/A

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

   12. lower-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. lower-fma.f64 N/A

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in v around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 97.4

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

8. Applied rewrites 97.4%

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

Alternative 5: 75.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e}{e + 1}\\ \mathbf{if}\;v \leq 0.00038:\\ \;\;\;\;v \cdot \mathsf{fma}\left(v \cdot v, t\_0 \cdot \mathsf{fma}\left(0.5, t\_0, -0.16666666666666666\right), t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;e \cdot \sin v\\ \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (let* ((t_0 (/ e (+ e 1.0))))
   (if (<= v 0.00038)
     (* v (fma (* v v) (* t_0 (fma 0.5 t_0 -0.16666666666666666)) t_0))
     (* e (sin v)))))

C

double code(double e, double v) {
	double t_0 = e / (e + 1.0);
	double tmp;
	if (v <= 0.00038) {
		tmp = v * fma((v * v), (t_0 * fma(0.5, t_0, -0.16666666666666666)), t_0);
	} else {
		tmp = e * sin(v);
	}
	return tmp;
}

Julia

function code(e, v)
	t_0 = Float64(e / Float64(e + 1.0))
	tmp = 0.0
	if (v <= 0.00038)
		tmp = Float64(v * fma(Float64(v * v), Float64(t_0 * fma(0.5, t_0, -0.16666666666666666)), t_0));
	else
		tmp = Float64(e * sin(v));
	end
	return tmp
end

Wolfram

code[e_, v_] := Block[{t$95$0 = N[(e / N[(e + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, 0.00038], N[(v * N[(N[(v * v), $MachinePrecision] * N[(t$95$0 * N[(0.5 * t$95$0 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < 3.8000000000000002e-4

   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

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 63.5%

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

   if 3.8000000000000002e-4 < v

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 97.4

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

   5. Applied rewrites 97.4%

      \[\leadsto \color{blue}{e \cdot \sin v} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 0.00038:\\ \;\;\;\;v \cdot \mathsf{fma}\left(v \cdot v, \frac{e}{e + 1} \cdot \mathsf{fma}\left(0.5, \frac{e}{e + 1}, -0.16666666666666666\right), \frac{e}{e + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;e \cdot \sin v\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 100:\\ \;\;\;\;v \cdot \mathsf{fma}\left(v \cdot v, \frac{e \cdot \left(v \cdot v\right)}{e + 1} \cdot \mathsf{fma}\left(v \cdot v, -0.0001984126984126984, 0.008333333333333333\right), \frac{e}{e + 1} \cdot \mathsf{fma}\left(v, v \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{1}{\mathsf{fma}\left(v \cdot v, 0.16666666666666666, 1\right) \cdot \frac{1}{e}}\\ \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (if (<= v 100.0)
   (*
    v
    (fma
     (* v v)
     (*
      (/ (* e (* v v)) (+ e 1.0))
      (fma (* v v) -0.0001984126984126984 0.008333333333333333))
     (* (/ e (+ e 1.0)) (fma v (* v -0.16666666666666666) 1.0))))
   (* v (/ 1.0 (* (fma (* v v) 0.16666666666666666 1.0) (/ 1.0 e))))))

C

double code(double e, double v) {
	double tmp;
	if (v <= 100.0) {
		tmp = v * fma((v * v), (((e * (v * v)) / (e + 1.0)) * fma((v * v), -0.0001984126984126984, 0.008333333333333333)), ((e / (e + 1.0)) * fma(v, (v * -0.16666666666666666), 1.0)));
	} else {
		tmp = v * (1.0 / (fma((v * v), 0.16666666666666666, 1.0) * (1.0 / e)));
	}
	return tmp;
}

Julia

function code(e, v)
	tmp = 0.0
	if (v <= 100.0)
		tmp = Float64(v * fma(Float64(v * v), Float64(Float64(Float64(e * Float64(v * v)) / Float64(e + 1.0)) * fma(Float64(v * v), -0.0001984126984126984, 0.008333333333333333)), Float64(Float64(e / Float64(e + 1.0)) * fma(v, Float64(v * -0.16666666666666666), 1.0))));
	else
		tmp = Float64(v * Float64(1.0 / Float64(fma(Float64(v * v), 0.16666666666666666, 1.0) * Float64(1.0 / e))));
	end
	return tmp
end

Wolfram

code[e_, v_] := If[LessEqual[v, 100.0], N[(v * N[(N[(v * v), $MachinePrecision] * N[(N[(N[(e * N[(v * v), $MachinePrecision]), $MachinePrecision] / N[(e + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(v * v), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[(e / N[(e + 1.0), $MachinePrecision]), $MachinePrecision] * N[(v * N[(v * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(v * N[(1.0 / N[(N[(N[(v * v), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[(1.0 / e), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 100

   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. lower-+.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in v around 0

      \[\leadsto \color{blue}{v \cdot \left({v}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{e}{1 + e} + {v}^{2} \cdot \left(\frac{-1}{5040} \cdot \frac{e \cdot {v}^{2}}{1 + e} + \frac{1}{120} \cdot \frac{e}{1 + e}\right)\right) + \frac{e}{1 + e}\right)} \]

   8. Applied rewrites 63.3%

      \[\leadsto \color{blue}{v \cdot \mathsf{fma}\left(v \cdot v, \frac{e \cdot \left(v \cdot v\right)}{e + 1} \cdot \mathsf{fma}\left(v \cdot v, -0.0001984126984126984, 0.008333333333333333\right), \mathsf{fma}\left(v, v \cdot -0.16666666666666666, 1\right) \cdot \frac{e}{e + 1}\right)} \]

   if 100 < v

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in v around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 2.5

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

   8. Applied rewrites 2.5%

      \[\leadsto \color{blue}{v \cdot \mathsf{fma}\left(e, v \cdot \left(v \cdot -0.16666666666666666\right), e\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. clear-num N/A

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

      7. flip-+ N/A

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

      8. lift-fma.f64 N/A

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

      9. lower-/.f64 2.5

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   10. Applied rewrites 2.5%

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

   11. Taylor expanded in v around 0

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

       1. associate-*r/ N/A

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

       2. *-rgt-identity N/A

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

       3. associate-*r/ N/A

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

       4. distribute-lft1-in N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 5.9

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

   13. Applied rewrites 5.9%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 100:\\ \;\;\;\;v \cdot \mathsf{fma}\left(v \cdot v, \frac{e \cdot \left(v \cdot v\right)}{e + 1} \cdot \mathsf{fma}\left(v \cdot v, -0.0001984126984126984, 0.008333333333333333\right), \frac{e}{e + 1} \cdot \mathsf{fma}\left(v, v \cdot -0.16666666666666666, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{1}{\mathsf{fma}\left(v \cdot v, 0.16666666666666666, 1\right) \cdot \frac{1}{e}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e}{e + 1}\\ \mathbf{if}\;v \leq 100:\\ \;\;\;\;v \cdot \mathsf{fma}\left(v \cdot v, t\_0 \cdot \mathsf{fma}\left(0.5, t\_0, -0.16666666666666666\right), t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{1}{\mathsf{fma}\left(v \cdot v, 0.16666666666666666, 1\right) \cdot \frac{1}{e}}\\ \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (let* ((t_0 (/ e (+ e 1.0))))
   (if (<= v 100.0)
     (* v (fma (* v v) (* t_0 (fma 0.5 t_0 -0.16666666666666666)) t_0))
     (* v (/ 1.0 (* (fma (* v v) 0.16666666666666666 1.0) (/ 1.0 e)))))))

C

double code(double e, double v) {
	double t_0 = e / (e + 1.0);
	double tmp;
	if (v <= 100.0) {
		tmp = v * fma((v * v), (t_0 * fma(0.5, t_0, -0.16666666666666666)), t_0);
	} else {
		tmp = v * (1.0 / (fma((v * v), 0.16666666666666666, 1.0) * (1.0 / e)));
	}
	return tmp;
}

Julia

function code(e, v)
	t_0 = Float64(e / Float64(e + 1.0))
	tmp = 0.0
	if (v <= 100.0)
		tmp = Float64(v * fma(Float64(v * v), Float64(t_0 * fma(0.5, t_0, -0.16666666666666666)), t_0));
	else
		tmp = Float64(v * Float64(1.0 / Float64(fma(Float64(v * v), 0.16666666666666666, 1.0) * Float64(1.0 / e))));
	end
	return tmp
end

Wolfram

code[e_, v_] := Block[{t$95$0 = N[(e / N[(e + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, 100.0], N[(v * N[(N[(v * v), $MachinePrecision] * N[(t$95$0 * N[(0.5 * t$95$0 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(v * N[(1.0 / N[(N[(N[(v * v), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] * N[(1.0 / e), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < 100

   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

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 63.1%

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

   if 100 < v

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in v around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 2.5

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

   8. Applied rewrites 2.5%

      \[\leadsto \color{blue}{v \cdot \mathsf{fma}\left(e, v \cdot \left(v \cdot -0.16666666666666666\right), e\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. clear-num N/A

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

      7. flip-+ N/A

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

      8. lift-fma.f64 N/A

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

      9. lower-/.f64 2.5

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

   10. Applied rewrites 2.5%

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

   11. Taylor expanded in v around 0

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

       1. associate-*r/ N/A

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

       2. *-rgt-identity N/A

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

       3. associate-*r/ N/A

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

       4. distribute-lft1-in N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 5.9

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

   13. Applied rewrites 5.9%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 100:\\ \;\;\;\;v \cdot \mathsf{fma}\left(v \cdot v, \frac{e}{e + 1} \cdot \mathsf{fma}\left(0.5, \frac{e}{e + 1}, -0.16666666666666666\right), \frac{e}{e + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;v \cdot \frac{1}{\mathsf{fma}\left(v \cdot v, 0.16666666666666666, 1\right) \cdot \frac{1}{e}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.2% accurate, 11.3× 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. Add Preprocessing

3. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 47.0

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

5. Applied rewrites 47.0%

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

6. Final simplification 47.0%

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

Alternative 9: 50.9% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 47.0

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

5. Applied rewrites 47.0%

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

6. Taylor expanded in e around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 46.3

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

8. Applied rewrites 46.3%

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

Alternative 10: 50.8% 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. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 47.0

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

5. Applied rewrites 47.0%

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

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. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

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

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

   6. unpow2 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-rgt-in N/A

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

   9. lower-*.f64 N/A

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

   10. mul-1-neg N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 46.0

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

8. Applied rewrites 46.0%

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

   1. cancel-sign-sub-inv N/A

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

   2. distribute-rgt1-in N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. lower-neg.f64 46.0

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

10. Applied rewrites 46.0%

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

11. Final simplification 46.0%

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

Alternative 11: 50.8% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 47.0

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

5. Applied rewrites 47.0%

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

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. distribute-lft-in N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

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

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

   6. unpow2 N/A

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

   7. mul-1-neg N/A

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

   8. distribute-rgt-in N/A

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

   9. lower-*.f64 N/A

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

   10. mul-1-neg N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 46.0

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

8. Applied rewrites 46.0%

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

Alternative 12: 50.2% 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 v around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 47.0

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

5. Applied rewrites 47.0%

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

6. Taylor expanded in e around 0

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

   1. lower-*.f64 45.3

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

8. Applied rewrites 45.3%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))