Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 5.2s

Alternatives: 12

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(N[Sin[v], $MachinePrecision] * e), $MachinePrecision] / N[(N[(N[Cos[v], $MachinePrecision] * e), $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. Final simplification 99.8%

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

Alternative 2: 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 3: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[e_, v_] := N[(N[(N[((-e) * N[Cos[v], $MachinePrecision] + 1.0), $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 v around inf

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

   1. rgt-mult-inverse N/A

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

   2. distribute-lft-in N/A

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

   3. +-commutative N/A

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

   4. times-frac N/A

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

   5. *-rgt-identity N/A

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

   6. associate-*r/ N/A

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

   7. rgt-mult-inverse N/A

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

   8. *-lft-identity N/A

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

   9. lower-/.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. metadata-eval N/A

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

   12. distribute-neg-frac N/A

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

   13. sub-neg N/A

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

   14. lower--.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-/.f64 99.6

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.3%

   \[\leadsto \frac{\sin v}{\mathsf{fma}\left({e}^{-0.5}, \color{blue}{{e}^{-0.5}}, \cos v\right)} \]
8. Step-by-step derivation

9. Applied rewrites 99.4%

   \[\leadsto \frac{\sin v}{\mathsf{fma}\left({e}^{-0.25}, \color{blue}{{e}^{-0.75}}, \cos v\right)} \]

10. Taylor expanded in e around 0

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

    1. +-commutative N/A

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

    2. associate-*r* N/A

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

    3. associate-*r* N/A

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

    4. *-lft-identity N/A

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

    5. distribute-rgt-in N/A

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

    6. +-commutative N/A

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

    7. *-commutative N/A

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

    8. associate-*r* N/A

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

    9. lower-*.f64 N/A

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

    10. lower-*.f64 N/A

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

    11. +-commutative N/A

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

    12. associate-*r* N/A

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

    13. lower-fma.f64 N/A

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

    14. mul-1-neg N/A

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

    15. lower-neg.f64 N/A

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

    16. lower-cos.f64 N/A

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

    17. lower-sin.f64 99.1

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

12. Applied rewrites 99.1%

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

13. Final simplification 99.1%

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

Alternative 4: 98.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(N[Sin[v], $MachinePrecision] * e), $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 98.6

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

5. Applied rewrites 98.6%

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

6. Final simplification 98.6%

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

Alternative 5: 98.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. lower-+.f64 98.6

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

5. Applied rewrites 98.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 98.6

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

7. Applied rewrites 98.6%

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

Alternative 6: 98.3% accurate, 2.0× speedup

Math

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

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(N[(1.0 - e), $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 v around inf

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

   1. rgt-mult-inverse N/A

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

   2. distribute-lft-in N/A

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

   3. +-commutative N/A

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

   4. times-frac N/A

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

   5. *-rgt-identity N/A

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

   6. associate-*r/ N/A

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

   7. rgt-mult-inverse N/A

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

   8. *-lft-identity N/A

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

   9. lower-/.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. metadata-eval N/A

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

   12. distribute-neg-frac N/A

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

   13. sub-neg N/A

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

   14. lower--.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-/.f64 99.6

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.3%

   \[\leadsto \frac{\sin v}{\mathsf{fma}\left({e}^{-0.5}, \color{blue}{{e}^{-0.5}}, \cos v\right)} \]

8. Taylor expanded in e around 0

   \[\leadsto \color{blue}{e \cdot \left(\sin v + -1 \cdot \left(e \cdot \left(\cos v \cdot \sin v\right)\right)\right)} \]
9. 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. associate-*r* N/A

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

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

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

   6. distribute-rgt1-in N/A

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

   7. lower-*.f64 N/A

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

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

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

   9. mul-1-neg N/A

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

   10. lower-fma.f64 N/A

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

   11. mul-1-neg N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-sin.f64 99.1

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

10. Applied rewrites 99.1%

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

11. Taylor expanded in v around 0

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

13. Applied rewrites 98.5%

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

Alternative 7: 97.7% 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 98.2

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

5. Applied rewrites 98.2%

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

Alternative 8: 52.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[e_, v_] := N[(1.0 / N[(N[(-0.3333333333333333 * v + N[(1.0 / v), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.16666666666666666 * v + N[(-1.0 / v), $MachinePrecision]), $MachinePrecision] / e), $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. 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 98.8

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

   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 98.8

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

4. Applied rewrites 98.8%

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

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 49.5%

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

8. Taylor expanded in e around -inf

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

10. Applied rewrites 49.5%

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

Alternative 9: 52.5% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[e_, v_] := N[(1.0 / N[(N[(-0.3333333333333333 * 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 98.8

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

   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 98.8

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

4. Applied rewrites 98.8%

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

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 49.5%

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

8. Taylor expanded in e around inf

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

10. Applied rewrites 49.2%

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

Alternative 10: 52.9% 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 49.2

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

5. Applied rewrites 49.2%

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

Alternative 11: 52.4% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(N[(1.0 - e), $MachinePrecision] * e), $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 49.2

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

5. Applied rewrites 49.2%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 49.1%

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

Alternative 12: 51.9% accurate, 37.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in 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 49.2

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

5. Applied rewrites 49.2%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 48.7%

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

9. Final simplification 48.7%

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

Reproduce

herbie shell --seed 2024304 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))