Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 7.7s

Alternatives: 10

Speedup: N/A×

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

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

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

Wolfram

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

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 99.8

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 99.8

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. mul-1-neg N/A

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

   3. associate-*r* N/A

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

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

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

   5. distribute-rgt1-in N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. mul-1-neg N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 99.3

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

5. Simplified 99.3%

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

6. Final simplification 99.3%

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

Alternative 3: 98.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 99.8

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 99.8

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in v around 0

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

Alternative 4: 98.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. mul-1-neg N/A

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

   3. associate-*r* N/A

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

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

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

   5. distribute-rgt1-in N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. mul-1-neg N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 99.3

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

5. Simplified 99.3%

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

6. Taylor expanded in v around 0

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

8. Simplified 98.8%

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

9. Final simplification 98.8%

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

Alternative 5: 97.8% 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. lower-*.f64 N/A

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

   2. lower-sin.f64 98.4

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

5. Simplified 98.4%

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

6. Final simplification 98.4%

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

Alternative 6: 52.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 2400000:\\ \;\;\;\;e \cdot \mathsf{fma}\left(v, \left(v \cdot v\right) \cdot \left(\frac{e}{\left(e + 1\right) \cdot \left(e + 1\right)} \cdot 0.5 + \frac{0.16666666666666666}{-1 - e}\right), \frac{v}{e + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;e \cdot \left(v \cdot \left(-e\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (if (<= v 2400000.0)
   (*
    e
    (fma
     v
     (*
      (* v v)
      (+
       (* (/ e (* (+ e 1.0) (+ e 1.0))) 0.5)
       (/ 0.16666666666666666 (- -1.0 e))))
     (/ v (+ e 1.0))))
   (* e (* v (- e)))))

C

double code(double e, double v) {
	double tmp;
	if (v <= 2400000.0) {
		tmp = e * fma(v, ((v * v) * (((e / ((e + 1.0) * (e + 1.0))) * 0.5) + (0.16666666666666666 / (-1.0 - e)))), (v / (e + 1.0)));
	} else {
		tmp = e * (v * -e);
	}
	return tmp;
}

Julia

function code(e, v)
	tmp = 0.0
	if (v <= 2400000.0)
		tmp = Float64(e * fma(v, Float64(Float64(v * v) * Float64(Float64(Float64(e / Float64(Float64(e + 1.0) * Float64(e + 1.0))) * 0.5) + Float64(0.16666666666666666 / Float64(-1.0 - e)))), Float64(v / Float64(e + 1.0))));
	else
		tmp = Float64(e * Float64(v * Float64(-e)));
	end
	return tmp
end

Wolfram

code[e_, v_] := If[LessEqual[v, 2400000.0], N[(e * N[(v * N[(N[(v * v), $MachinePrecision] * N[(N[(N[(e / N[(N[(e + 1.0), $MachinePrecision] * N[(e + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + N[(0.16666666666666666 / N[(-1.0 - e), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(v / N[(e + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(e * N[(v * (-e)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 2.4e6

   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-sin.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 99.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in v around 0

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

      1. distribute-lft-in N/A

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

      2. lower-fma.f64 N/A

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

   7. Simplified 68.3%

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

   if 2.4e6 < v

   1. Initial program 99.6%

      \[\frac{e \cdot \sin v}{1 + e \cdot \cos v} \]
   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. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

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

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

      5. distribute-rgt1-in N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-sin.f64 99.4

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

   5. Simplified 99.4%

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

   6. Taylor expanded in v around 0

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. mul-1-neg N/A

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

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

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 3.3

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

   8. Simplified 3.3%

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

   9. Taylor expanded in e around inf

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

       1. *-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. mul-1-neg N/A

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

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

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

       9. mul-1-neg N/A

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

       10. lower-*.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lower-neg.f64 5.9

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

   11. Simplified 5.9%

       \[\leadsto \color{blue}{e \cdot \left(e \cdot \left(-v\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 2400000:\\ \;\;\;\;e \cdot \mathsf{fma}\left(v, \left(v \cdot v\right) \cdot \left(\frac{e}{\left(e + 1\right) \cdot \left(e + 1\right)} \cdot 0.5 + \frac{0.16666666666666666}{-1 - e}\right), \frac{v}{e + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;e \cdot \left(v \cdot \left(-e\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.8% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-*r/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 50.0

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

5. Simplified 50.0%

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

   1. lift-+.f64 N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. associate-/r/ N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 50.0

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

7. Applied egg-rr 50.0%

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

8. Final simplification 50.0%

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

Alternative 8: 51.8% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-*r/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 50.0

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

5. Simplified 50.0%

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

Alternative 9: 51.4% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(e * N[(v - N[(v * 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 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. lower-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. associate-*r* N/A

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

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

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

   5. distribute-rgt1-in N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. mul-1-neg N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 99.3

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

5. Simplified 99.3%

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

6. Taylor expanded in v around 0

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. mul-1-neg N/A

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

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

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 49.7

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

8. Simplified 49.7%

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

9. Final simplification 49.7%

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

Alternative 10: 50.9% 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. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 50.0

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

5. Simplified 50.0%

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

6. Taylor expanded in e around 0

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

   1. lower-*.f64 49.3

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

8. Simplified 49.3%

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

9. Final simplification 49.3%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))