Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 7.6s

Alternatives: 14

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

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

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

Wolfram

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

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 99.8

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 99.8

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in e around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 99.5

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.5%

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

Alternative 3: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 99.8

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in e around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 99.5

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

7. Applied rewrites 99.5%

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

Alternative 4: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. mul-1-neg N/A

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

   4. associate-*r* N/A

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

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

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt1-in N/A

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

   9. lower-*.f64 N/A

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

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

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

   11. mul-1-neg N/A

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

   12. lower-fma.f64 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f64 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-cos.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-sin.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Final simplification 99.4%

   \[\leadsto \left(\sin v \cdot e\right) \cdot \mathsf{fma}\left(-e, \cos v, 1\right) \]
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. 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. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 99.8

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-sin.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in v around 0

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

   1. lower-+.f64 98.9

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

9. Applied rewrites 98.9%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. *-commutative N/A

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

    4. associate-/l* N/A

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

    5. lower-*.f64 N/A

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

    6. lower-/.f64 98.9

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

11. Applied rewrites 98.9%

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

12. Final simplification 98.9%

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

Alternative 6: 98.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in e around 0

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

   1. distribute-lft-in N/A

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

   2. *-commutative N/A

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

   3. mul-1-neg N/A

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

   4. associate-*r* N/A

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

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

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. distribute-rgt1-in N/A

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

   9. lower-*.f64 N/A

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

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

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

   11. mul-1-neg N/A

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

   12. lower-fma.f64 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f64 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-cos.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-sin.f64 99.4

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

5. Applied rewrites 99.4%

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

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 98.9%

   \[\leadsto \left(1 - e\right) \cdot \left(\color{blue}{\sin v} \cdot e\right) \]
9. 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 98.6

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

5. Applied rewrites 98.6%

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

Alternative 8: 51.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.5, e, \mathsf{fma}\left(0.16666666666666666, e, 0.16666666666666666\right)\right)\\ \frac{e}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, e, \mathsf{fma}\left(t\_0, 0.16666666666666666, \mathsf{fma}\left(-0.008333333333333333, e, -0.008333333333333333\right)\right)\right), v \cdot v, t\_0\right), v \cdot v, 1 + e\right)}{v}} \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (let* ((t_0 (fma -0.5 e (fma 0.16666666666666666 e 0.16666666666666666))))
   (/
    e
    (/
     (fma
      (fma
       (fma
        0.041666666666666664
        e
        (fma
         t_0
         0.16666666666666666
         (fma -0.008333333333333333 e -0.008333333333333333)))
       (* v v)
       t_0)
      (* v v)
      (+ 1.0 e))
     v))))

C

double code(double e, double v) {
	double t_0 = fma(-0.5, e, fma(0.16666666666666666, e, 0.16666666666666666));
	return e / (fma(fma(fma(0.041666666666666664, e, fma(t_0, 0.16666666666666666, fma(-0.008333333333333333, e, -0.008333333333333333))), (v * v), t_0), (v * v), (1.0 + e)) / v);
}

Julia

function code(e, v)
	t_0 = fma(-0.5, e, fma(0.16666666666666666, e, 0.16666666666666666))
	return Float64(e / Float64(fma(fma(fma(0.041666666666666664, e, fma(t_0, 0.16666666666666666, fma(-0.008333333333333333, e, -0.008333333333333333))), Float64(v * v), t_0), Float64(v * v), Float64(1.0 + e)) / v))
end

Wolfram

code[e_, v_] := Block[{t$95$0 = N[(-0.5 * e + N[(0.16666666666666666 * e + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, N[(e / N[(N[(N[(N[(0.041666666666666664 * e + N[(t$95$0 * 0.16666666666666666 + N[(-0.008333333333333333 * e + -0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(v * v), $MachinePrecision] + t$95$0), $MachinePrecision] * N[(v * v), $MachinePrecision] + N[(1.0 + e), $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. 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. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.7

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in v around 0

   \[\leadsto \frac{e}{\color{blue}{\frac{1 + \left(e + {v}^{2} \cdot \left(\left(\frac{-1}{2} \cdot e + {v}^{2} \cdot \left(\frac{1}{24} \cdot e - \left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right) + \frac{1}{120} \cdot \left(1 + e\right)\right)\right)\right) - \frac{-1}{6} \cdot \left(1 + e\right)\right)\right)}{v}}} \]

7. Applied rewrites 55.4%

   \[\leadsto \frac{e}{\color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, e, \mathsf{fma}\left(\mathsf{fma}\left(-0.5, e, \mathsf{fma}\left(0.16666666666666666, e, 0.16666666666666666\right)\right), 0.16666666666666666, \mathsf{fma}\left(-0.008333333333333333, e, -0.008333333333333333\right)\right)\right), v \cdot v, \mathsf{fma}\left(-0.5, e, \mathsf{fma}\left(0.16666666666666666, e, 0.16666666666666666\right)\right)\right), v \cdot v, 1 + e\right)}{v}}} \]
8. Add Preprocessing

Alternative 9: 51.7% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.7

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in v around 0

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

   1. lower-/.f64 N/A

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

   2. associate-+r+ N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. lower-fma.f64 N/A

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

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

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

   11. distribute-lft-in N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-+.f64 55.4

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

7. Applied rewrites 55.4%

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

Alternative 10: 50.6% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 54.2

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

5. Applied rewrites 54.2%

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

6. Final simplification 54.2%

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

Alternative 11: 50.2% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 54.2

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

5. Applied rewrites 54.2%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 54.2%

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

Alternative 12: 50.0% accurate, 11.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in 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 54.2

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

5. Applied rewrites 54.2%

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

7. Applied rewrites 54.1%

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

8. Taylor expanded in e around 0

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

10. Applied rewrites 54.2%

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

12. Applied rewrites 54.2%

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

Alternative 13: 50.0% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[e_, v_] := N[(N[(v - N[(v * 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

   \[\leadsto \color{blue}{\frac{e \cdot v}{1 + e}} \]
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 54.2

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

5. Applied rewrites 54.2%

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

7. Applied rewrites 54.1%

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

8. Taylor expanded in e around 0

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

10. Applied rewrites 54.2%

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

Alternative 14: 49.4% 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. +-commutative N/A

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

   5. lower-+.f64 54.2

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

5. Applied rewrites 54.2%

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

6. Taylor expanded in e around 0

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

8. Applied rewrites 53.9%

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

Reproduce

herbie shell --seed 2024235 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))