Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 6.3s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[e_, v_] := N[(N[(N[Sin[v], $MachinePrecision] / N[(N[Cos[v], $MachinePrecision] * e + 1.0), $MachinePrecision]), $MachinePrecision] * e), $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-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.8

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 98.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Applied rewrites 98.2%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 98.2

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

7. Applied rewrites 98.2%

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

Alternative 3: 98.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(N[(1.0 - e), $MachinePrecision] * e), $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. 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

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

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

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

   5. associate-*r* N/A

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

   6. distribute-rgt1-in N/A

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

   7. lower-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-sin.f64 97.9

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

5. Applied rewrites 97.9%

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

6. Taylor expanded in v around inf

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

8. Applied rewrites 97.9%

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

9. Taylor expanded in v around 0

   \[\leadsto \left(e \cdot \left(1 + -1 \cdot e\right)\right) \cdot \sin v \]
10. Step-by-step derivation

11. Applied rewrites 97.1%

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

Alternative 4: 75.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{e}{e - -1} \cdot v\\ \mathbf{else}:\\ \;\;\;\;\sin v \cdot e\\ \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (if (<= v 9.5e-12) (* (/ e (- e -1.0)) v) (* (sin v) e)))

C

double code(double e, double v) {
	double tmp;
	if (v <= 9.5e-12) {
		tmp = (e / (e - -1.0)) * v;
	} else {
		tmp = sin(v) * e;
	}
	return tmp;
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    real(8) :: tmp
    if (v <= 9.5d-12) then
        tmp = (e / (e - (-1.0d0))) * v
    else
        tmp = sin(v) * e
    end if
    code = tmp
end function

Java

public static double code(double e, double v) {
	double tmp;
	if (v <= 9.5e-12) {
		tmp = (e / (e - -1.0)) * v;
	} else {
		tmp = Math.sin(v) * e;
	}
	return tmp;
}

Python

def code(e, v):
	tmp = 0
	if v <= 9.5e-12:
		tmp = (e / (e - -1.0)) * v
	else:
		tmp = math.sin(v) * e
	return tmp

Julia

function code(e, v)
	tmp = 0.0
	if (v <= 9.5e-12)
		tmp = Float64(Float64(e / Float64(e - -1.0)) * v);
	else
		tmp = Float64(sin(v) * e);
	end
	return tmp
end

MATLAB

function tmp_2 = code(e, v)
	tmp = 0.0;
	if (v <= 9.5e-12)
		tmp = (e / (e - -1.0)) * v;
	else
		tmp = sin(v) * e;
	end
	tmp_2 = tmp;
end

Wolfram

code[e_, v_] := If[LessEqual[v, 9.5e-12], N[(N[(e / N[(e - -1.0), $MachinePrecision]), $MachinePrecision] * v), $MachinePrecision], N[(N[Sin[v], $MachinePrecision] * e), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 9.4999999999999995e-12

   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. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 60.6

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

   5. Applied rewrites 60.6%

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

   if 9.4999999999999995e-12 < v

   1. Initial program 99.5%

      \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.5

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in e around 0

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

      1. lower-sin.f64 99.5

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

   7. Applied rewrites 99.5%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 9.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{e}{e - -1} \cdot v\\ \mathbf{else}:\\ \;\;\;\;\sin v \cdot e\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1 + \mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, e, 0.16666666666666666\right), v \cdot v, e\right)}{v}} \cdot e \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (*
  (/
   1.0
   (/
    (+ 1.0 (fma (fma -0.3333333333333333 e 0.16666666666666666) (* v v) e))
    v))
  e))

C

double code(double e, double v) {
	return (1.0 / ((1.0 + fma(fma(-0.3333333333333333, e, 0.16666666666666666), (v * v), e)) / v)) * e;
}

Julia

function code(e, v)
	return Float64(Float64(1.0 / Float64(Float64(1.0 + fma(fma(-0.3333333333333333, e, 0.16666666666666666), Float64(v * v), e)) / v)) * e)
end

Wolfram

code[e_, v_] := N[(N[(1.0 / N[(N[(1.0 + N[(N[(-0.3333333333333333 * e + 0.16666666666666666), $MachinePrecision] * N[(v * v), $MachinePrecision] + e), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision] * e), $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-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.8

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lift-/.f64 N/A

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

   4. lower-/.f64 99.6

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

   5. lift-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in v around 0

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

   1. div-add N/A

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

   2. *-commutative N/A

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

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

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. metadata-eval N/A

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

   8. associate-+l+ N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. div-add N/A

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

   12. lower-/.f64 N/A

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

9. Applied rewrites 45.4%

   \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, e, 0.16666666666666666\right), v \cdot v, e\right) + 1}{v}}} \cdot e \]

10. Final simplification 45.4%

    \[\leadsto \frac{1}{\frac{1 + \mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, e, 0.16666666666666666\right), v \cdot v, e\right)}{v}} \cdot e \]
11. Add Preprocessing

Alternative 6: 52.8% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \frac{e}{\mathsf{fma}\left(0.16666666666666666, v, \mathsf{fma}\left(-0.3333333333333333 \cdot v, e, \frac{e - -1}{v}\right)\right)} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (/
  e
  (fma
   0.16666666666666666
   v
   (fma (* -0.3333333333333333 v) e (/ (- e -1.0) v)))))

C

double code(double e, double v) {
	return e / fma(0.16666666666666666, v, fma((-0.3333333333333333 * v), e, ((e - -1.0) / v)));
}

Julia

function code(e, v)
	return Float64(e / fma(0.16666666666666666, v, fma(Float64(-0.3333333333333333 * v), e, Float64(Float64(e - -1.0) / v))))
end

Wolfram

code[e_, v_] := N[(e / N[(0.16666666666666666 * v + N[(N[(-0.3333333333333333 * v), $MachinePrecision] * e + N[(N[(e - -1.0), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.6

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. associate-+r+ N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. lower-fma.f64 N/A

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

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

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

   11. distribute-lft-in N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-+.f64 45.4

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

7. Applied rewrites 45.4%

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

8. Taylor expanded in e around 0

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

10. Applied rewrites 45.4%

    \[\leadsto \frac{e}{\mathsf{fma}\left(0.16666666666666666, \color{blue}{v}, \mathsf{fma}\left(-0.3333333333333333 \cdot v, e, \frac{1 + e}{v}\right)\right)} \]

11. Final simplification 45.4%

    \[\leadsto \frac{e}{\mathsf{fma}\left(0.16666666666666666, v, \mathsf{fma}\left(-0.3333333333333333 \cdot v, e, \frac{e - -1}{v}\right)\right)} \]
12. Add Preprocessing

Alternative 7: 51.6% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(e / N[(e - -1.0), $MachinePrecision]), $MachinePrecision] * 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. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 44.4

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

5. Applied rewrites 44.4%

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

6. Final simplification 44.4%

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

Alternative 8: 51.1% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

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

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

   5. associate-*r* N/A

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

   6. distribute-rgt1-in N/A

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

   7. lower-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-sin.f64 97.9

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

5. Applied rewrites 97.9%

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

6. Taylor expanded in v around 0

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

8. Applied rewrites 43.3%

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

Alternative 9: 50.6% 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. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 44.4

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

5. Applied rewrites 44.4%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 42.1%

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

Reproduce

herbie shell --seed 2024312 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))