Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 7.7s

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: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[e_, v_] := N[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] * N[((-e) * N[Cos[v], $MachinePrecision] + 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 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 99.3

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

5. Applied rewrites 99.3%

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

6. Final simplification 99.3%

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

Alternative 3: 98.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. +-commutative N/A

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

   2. lower-+.f64 99.2

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

5. Applied rewrites 99.2%

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

7. Applied rewrites 99.2%

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

8. Final simplification 99.2%

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

Alternative 4: 98.7% 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. +-commutative N/A

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

   2. lower-+.f64 99.2

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

5. Applied rewrites 99.2%

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

6. Final simplification 99.2%

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

Alternative 5: 98.3% accurate, 2.0× speedup

Math

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

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(1.0 - e) * Float64(e * sin(v)))
end

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(1.0 - e), $MachinePrecision] * N[(e * N[Sin[v], $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. 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 99.3

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

5. Applied rewrites 99.3%

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

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 98.8%

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

9. Final simplification 98.8%

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

Alternative 6: 97.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in e around 0

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

   1. *-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 98.4

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

5. Applied rewrites 98.4%

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

6. Final simplification 98.4%

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

Alternative 7: 52.4% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[e_, v_] := N[(1.0 / N[(N[(N[(N[(N[(1.0 + e), $MachinePrecision] / e), $MachinePrecision] * 0.16666666666666666 + -0.5), $MachinePrecision] * N[(v * v), $MachinePrecision] + 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. lower-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.0

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.0

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 42.5%

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

8. Final simplification 42.5%

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

Alternative 8: 52.3% 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. +-commutative N/A

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

   5. lower-+.f64 41.9

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

5. Applied rewrites 41.9%

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

6. Final simplification 41.9%

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

Alternative 9: 51.8% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(v - N[(e * 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 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. +-commutative N/A

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

   5. lower-+.f64 41.9

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

5. Applied rewrites 41.9%

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

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 41.5%

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

Alternative 10: 51.3% 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. +-commutative N/A

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

   5. lower-+.f64 41.9

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

5. Applied rewrites 41.9%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 41.1%

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

Alternative 11: 3.8% accurate, 75.0× speedup

Math

\[\begin{array}{l} \\ -v \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(e, v):
	return -v

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := (-v)

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around 0

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

   1. 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. +-commutative N/A

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

   5. lower-+.f64 41.9

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

5. Applied rewrites 41.9%

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

7. Applied rewrites 41.3%

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

9. Applied rewrites 40.9%

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

10. Taylor expanded in e around inf

    \[\leadsto -1 \cdot \color{blue}{v} \]
11. Step-by-step derivation

12. Applied rewrites 3.4%

    \[\leadsto -v \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024278 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))