Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 7.2s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

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: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[e_, v_] := N[(N[(N[(N[Cos[v], $MachinePrecision] * (-e) + 1.0), $MachinePrecision] * N[Sin[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. Taylor expanded in e around 0

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

   1. distribute-lft-in N/A

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

   2. mul-1-neg N/A

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

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

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. associate-*r* N/A

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

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

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

   9. distribute-rgt1-in N/A

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

   10. lower-*.f64 N/A

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

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

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

   12. mul-1-neg N/A

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

   13. lower-fma.f64 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lower-sin.f64 98.8

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

5. Applied rewrites 98.8%

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

7. Applied rewrites 98.9%

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

Alternative 3: 98.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around 0

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

   1. lower-+.f64 98.8

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

5. Applied rewrites 98.8%

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

Alternative 4: 98.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(1.0 - e), $MachinePrecision] * N[(N[Sin[v], $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. distribute-lft-in N/A

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

   2. mul-1-neg N/A

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

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

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. associate-*r* N/A

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

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

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

   9. distribute-rgt1-in N/A

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

   10. lower-*.f64 N/A

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

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

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

   12. mul-1-neg N/A

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

   13. lower-fma.f64 N/A

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

   14. mul-1-neg N/A

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

   15. lower-neg.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. lower-sin.f64 98.8

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

5. Applied rewrites 98.8%

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

6. Taylor expanded in v around 0

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

8. Applied rewrites 98.3%

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

Alternative 5: 97.9% 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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 98.1

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

5. Applied rewrites 98.1%

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

Alternative 6: 52.7% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 N/A

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

   6. distribute-neg-frac N/A

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

   7. lower-/.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. distribute-neg-in N/A

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

   10. metadata-eval N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 99.1

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.1

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.1

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

4. Applied rewrites 99.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift--.f64 N/A

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

   5. div-sub N/A

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

   6. sub-div N/A

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

   7. frac-sub N/A

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

   8. lower-/.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 72.6

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

6. Applied rewrites 72.6%

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

7. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

9. Applied rewrites 46.6%

   \[\leadsto \frac{-1}{\frac{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(e \cdot e, 0.3333333333333333, -0.16666666666666666 \cdot e\right) \cdot v, v, e \cdot \left(-1 - e\right)\right)}{v}}}{e \cdot e}} \]
10. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \color{blue}{\frac{-1}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(e \cdot e, \frac{1}{3}, \frac{-1}{6} \cdot e\right) \cdot v, v, e \cdot \left(-1 - e\right)\right)}{v}}{e \cdot e}}} \]

    2. lift-/.f64 N/A

       \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(e \cdot e, \frac{1}{3}, \frac{-1}{6} \cdot e\right) \cdot v, v, e \cdot \left(-1 - e\right)\right)}{v}}{e \cdot e}}} \]

    3. associate-/r/ N/A

       \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(e \cdot e, \frac{1}{3}, \frac{-1}{6} \cdot e\right) \cdot v, v, e \cdot \left(-1 - e\right)\right)}{v}} \cdot \left(e \cdot e\right)} \]

    4. lift-*.f64 N/A

       \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(e \cdot e, \frac{1}{3}, \frac{-1}{6} \cdot e\right) \cdot v, v, e \cdot \left(-1 - e\right)\right)}{v}} \cdot \color{blue}{\left(e \cdot e\right)} \]

    5. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\frac{-1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(e \cdot e, \frac{1}{3}, \frac{-1}{6} \cdot e\right) \cdot v, v, e \cdot \left(-1 - e\right)\right)}{v}} \cdot e\right) \cdot e} \]

    6. lower-*.f64 N/A

       \[\leadsto \color{blue}{\left(\frac{-1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(e \cdot e, \frac{1}{3}, \frac{-1}{6} \cdot e\right) \cdot v, v, e \cdot \left(-1 - e\right)\right)}{v}} \cdot e\right) \cdot e} \]

11. Applied rewrites 52.0%

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

Alternative 7: 51.9% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. clear-num N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 N/A

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

   6. distribute-neg-frac N/A

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

   7. lower-/.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. distribute-neg-in N/A

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

   10. metadata-eval N/A

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

   11. unsub-neg N/A

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

   12. lower--.f64 99.1

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.1

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.1

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

4. Applied rewrites 99.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift--.f64 N/A

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

   5. div-sub N/A

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

   6. sub-div N/A

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

   7. frac-sub N/A

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

   8. lower-/.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 72.6

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

6. Applied rewrites 72.6%

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

7. Taylor expanded in v around 0

   \[\leadsto \frac{-1}{\color{blue}{\frac{\left({v}^{2} \cdot \left(\frac{1}{3} - \frac{1}{6} \cdot \frac{1}{e}\right) + \frac{-1}{e}\right) - 1}{v}}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{-1}{\color{blue}{\frac{\left({v}^{2} \cdot \left(\frac{1}{3} - \frac{1}{6} \cdot \frac{1}{e}\right) + \frac{-1}{e}\right) - 1}{v}}} \]

   2. associate--l+ N/A

      \[\leadsto \frac{-1}{\frac{\color{blue}{{v}^{2} \cdot \left(\frac{1}{3} - \frac{1}{6} \cdot \frac{1}{e}\right) + \left(\frac{-1}{e} - 1\right)}}{v}} \]

   3. *-commutative N/A

      \[\leadsto \frac{-1}{\frac{\color{blue}{\left(\frac{1}{3} - \frac{1}{6} \cdot \frac{1}{e}\right) \cdot {v}^{2}} + \left(\frac{-1}{e} - 1\right)}{v}} \]

   4. unpow2 N/A

      \[\leadsto \frac{-1}{\frac{\left(\frac{1}{3} - \frac{1}{6} \cdot \frac{1}{e}\right) \cdot \color{blue}{\left(v \cdot v\right)} + \left(\frac{-1}{e} - 1\right)}{v}} \]

   5. associate-*r* N/A

      \[\leadsto \frac{-1}{\frac{\color{blue}{\left(\left(\frac{1}{3} - \frac{1}{6} \cdot \frac{1}{e}\right) \cdot v\right) \cdot v} + \left(\frac{-1}{e} - 1\right)}{v}} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{-1}{\frac{\color{blue}{\mathsf{fma}\left(\left(\frac{1}{3} - \frac{1}{6} \cdot \frac{1}{e}\right) \cdot v, v, \frac{-1}{e} - 1\right)}}{v}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3} - \frac{1}{6} \cdot \frac{1}{e}\right) \cdot v}, v, \frac{-1}{e} - 1\right)}{v}} \]

   8. lower--.f64 N/A

      \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3} - \frac{1}{6} \cdot \frac{1}{e}\right)} \cdot v, v, \frac{-1}{e} - 1\right)}{v}} \]

   9. associate-*r/ N/A

      \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(\left(\frac{1}{3} - \color{blue}{\frac{\frac{1}{6} \cdot 1}{e}}\right) \cdot v, v, \frac{-1}{e} - 1\right)}{v}} \]

   10. metadata-eval N/A

       \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(\left(\frac{1}{3} - \frac{\color{blue}{\frac{1}{6}}}{e}\right) \cdot v, v, \frac{-1}{e} - 1\right)}{v}} \]

   11. lower-/.f64 N/A

       \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(\left(\frac{1}{3} - \color{blue}{\frac{\frac{1}{6}}{e}}\right) \cdot v, v, \frac{-1}{e} - 1\right)}{v}} \]

   12. lower--.f64 N/A

       \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(\left(\frac{1}{3} - \frac{\frac{1}{6}}{e}\right) \cdot v, v, \color{blue}{\frac{-1}{e} - 1}\right)}{v}} \]

   13. lower-/.f64 51.3

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

9. Applied rewrites 51.3%

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

Alternative 8: 51.6% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(e / N[(1.0 + e), $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 51.0

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

5. Applied rewrites 51.0%

   \[\leadsto \color{blue}{\frac{e}{1 + e} \cdot v} \]
6. Add Preprocessing

Alternative 9: 51.1% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[e_, v_] := N[(N[((-v) * e), $MachinePrecision] * e + N[(v * 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 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 51.0

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

5. Applied rewrites 51.0%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 50.6%

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

10. Applied rewrites 50.6%

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

Alternative 10: 51.1% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[e_, v_] := N[(N[((-v) * e + 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 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 51.0

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

5. Applied rewrites 51.0%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 50.6%

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

Alternative 11: 50.7% accurate, 37.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in 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 51.0

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

5. Applied rewrites 51.0%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 50.3%

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

Reproduce

herbie shell --seed 2024313 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))