Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 6.1s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.8

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.8

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in e around 0

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

   1. mul-1-neg N/A

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

   2. associate-*r* N/A

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

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

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

   4. *-lft-identity N/A

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

   5. mul-1-neg N/A

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

   6. distribute-rgt-in N/A

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

   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

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

7. Applied rewrites 99.6%

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

Alternative 3: 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. lower-+.f64 99.6

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

5. Applied rewrites 99.6%

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

Alternative 4: 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 98.7

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

5. Applied rewrites 98.7%

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

6. Final simplification 98.7%

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

Alternative 5: 51.4% 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 50.1

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

5. Applied rewrites 50.1%

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

6. Final simplification 50.1%

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

Alternative 6: 51.4% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around 0

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

   1. 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 50.1

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

5. Applied rewrites 50.1%

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

7. Applied rewrites 50.0%

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

8. Final simplification 50.0%

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

Alternative 7: 50.9% 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 50.1

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

5. Applied rewrites 50.1%

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

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 49.8%

   \[\leadsto \left(\left(1 - e\right) \cdot e\right) \cdot v \]

9. Final simplification 49.8%

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

Alternative 8: 50.6% accurate, 37.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-+.f64 50.1

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

5. Applied rewrites 50.1%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 49.2%

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

9. Final simplification 49.2%

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

Reproduce

herbie shell --seed 2024337 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))