Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 10.5s

Alternatives: 13

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, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{e \cdot \sin v}{e^{\mathsf{log1p}\left(e \cdot \cos v\right)}} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (/ (* e (sin v)) (exp (log1p (* e (cos v))))))

C

double code(double e, double v) {
	return (e * sin(v)) / exp(log1p((e * cos(v))));
}

Java

public static double code(double e, double v) {
	return (e * Math.sin(v)) / Math.exp(Math.log1p((e * Math.cos(v))));
}

Python

def code(e, v):
	return (e * math.sin(v)) / math.exp(math.log1p((e * math.cos(v))))

Julia

function code(e, v)
	return Float64(Float64(e * sin(v)) / exp(log1p(Float64(e * cos(v)))))
end

Wolfram

code[e_, v_] := N[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / N[Exp[N[Log[1 + N[(e * N[Cos[v], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

   1. /-rgt-identity N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \left(\frac{1 + e \cdot \cos v}{\color{blue}{1}}\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \left(\frac{1}{\color{blue}{\frac{1}{1 + e \cdot \cos v}}}\right)\right) \]

   3. inv-pow N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \left(\frac{1}{{\left(1 + e \cdot \cos v\right)}^{\color{blue}{-1}}}\right)\right) \]

   4. pow-to-exp N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \left(\frac{1}{e^{\log \left(1 + e \cdot \cos v\right) \cdot -1}}\right)\right) \]

   5. rec-exp N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \left(e^{\mathsf{neg}\left(\log \left(1 + e \cdot \cos v\right) \cdot -1\right)}\right)\right) \]

   6. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\log \left(1 + e \cdot \cos v\right) \cdot -1\right)\right)\right)\right) \]

   7. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\left(\log \left(1 + e \cdot \cos v\right) \cdot -1\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\log \left(1 + e \cdot \cos v\right), -1\right)\right)\right)\right) \]

   9. accelerator-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{log1p.f64}\left(\left(e \cdot \cos v\right)\right), -1\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(e, \cos v\right)\right), -1\right)\right)\right)\right) \]

   11. cos-lowering-cos.f64 99.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \mathsf{exp.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(e, \mathsf{cos.f64}\left(v\right)\right)\right), -1\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto \frac{e \cdot \sin v}{\color{blue}{e^{-\mathsf{log1p}\left(e \cdot \cos v\right) \cdot -1}}} \]

5. Final simplification 99.8%

   \[\leadsto \frac{e \cdot \sin v}{e^{\mathsf{log1p}\left(e \cdot \cos v\right)}} \]
6. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-/l* N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(e, \color{blue}{\left(\frac{1 + e \cdot \cos v}{\sin v}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\left(1 + e \cdot \cos v\right), \color{blue}{\sin v}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(e \cdot \cos v\right)\right), \sin \color{blue}{v}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(e, \cos v\right)\right), \sin v\right)\right) \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(e, \mathsf{cos.f64}\left(v\right)\right)\right), \sin v\right)\right) \]

   9. sin-lowering-sin.f64 99.6%

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(e, \mathsf{cos.f64}\left(v\right)\right)\right), \mathsf{sin.f64}\left(v\right)\right)\right) \]

4. Applied egg-rr 99.6%

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

5. Final simplification 99.6%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in v around inf

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

   1. rgt-mult-inverse N/A

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

   2. distribute-lft-in N/A

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

   3. +-commutative N/A

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

   4. times-frac N/A

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

   5. *-rgt-identity N/A

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

   6. associate-*r/ N/A

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

   7. rgt-mult-inverse N/A

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

   8. *-lft-identity N/A

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

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\sin v, \color{blue}{\left(\cos v + \frac{1}{e}\right)}\right) \]

   10. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(v\right), \left(\color{blue}{\cos v} + \frac{1}{e}\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(v\right), \mathsf{+.f64}\left(\cos v, \color{blue}{\left(\frac{1}{e}\right)}\right)\right) \]

   12. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(v\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(v\right), \left(\frac{\color{blue}{1}}{e}\right)\right)\right) \]

   13. /-lowering-/.f64 99.6%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(v\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(v\right), \mathsf{/.f64}\left(1, \color{blue}{e}\right)\right)\right) \]

5. Simplified 99.6%

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

Alternative 5: 98.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double e, double v) {
	return ((e * sin(v)) / (1.0 + (e * (e * e)))) * ((e * e) + (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 * (e * e)))) * ((e * e) + (1.0d0 - e))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in v around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \color{blue}{\left(1 + e\right)}\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \left(e + \color{blue}{1}\right)\right) \]

   2. +-lowering-+.f64 99.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \mathsf{+.f64}\left(e, \color{blue}{1}\right)\right) \]

5. Simplified 99.0%

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

   1. flip3-+ N/A

      \[\leadsto \frac{e \cdot \sin v}{\frac{{e}^{3} + {1}^{3}}{\color{blue}{e \cdot e + \left(1 \cdot 1 - e \cdot 1\right)}}} \]

   2. associate-/r/ N/A

      \[\leadsto \frac{e \cdot \sin v}{{e}^{3} + {1}^{3}} \cdot \color{blue}{\left(e \cdot e + \left(1 \cdot 1 - e \cdot 1\right)\right)} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{e \cdot \sin v}{{e}^{3} + {1}^{3}}\right), \color{blue}{\left(e \cdot e + \left(1 \cdot 1 - e \cdot 1\right)\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(e \cdot \sin v\right), \left({e}^{3} + {1}^{3}\right)\right), \left(\color{blue}{e \cdot e} + \left(1 \cdot 1 - e \cdot 1\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \sin v\right), \left({e}^{3} + {1}^{3}\right)\right), \left(\color{blue}{e} \cdot e + \left(1 \cdot 1 - e \cdot 1\right)\right)\right) \]

   6. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \left({e}^{3} + {1}^{3}\right)\right), \left(e \cdot e + \left(1 \cdot 1 - e \cdot 1\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \left({e}^{3} + 1\right)\right), \left(e \cdot e + \left(1 \cdot 1 - e \cdot 1\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \left(1 + {e}^{3}\right)\right), \left(e \cdot \color{blue}{e} + \left(1 \cdot 1 - e \cdot 1\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \mathsf{+.f64}\left(1, \left({e}^{3}\right)\right)\right), \left(e \cdot \color{blue}{e} + \left(1 \cdot 1 - e \cdot 1\right)\right)\right) \]

   10. cube-mult N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \mathsf{+.f64}\left(1, \left(e \cdot \left(e \cdot e\right)\right)\right)\right), \left(e \cdot e + \left(1 \cdot 1 - e \cdot 1\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(e, \left(e \cdot e\right)\right)\right)\right), \left(e \cdot e + \left(1 \cdot 1 - e \cdot 1\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(e, \mathsf{*.f64}\left(e, e\right)\right)\right)\right), \left(e \cdot e + \left(1 \cdot 1 - e \cdot 1\right)\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(e, \mathsf{*.f64}\left(e, e\right)\right)\right)\right), \mathsf{+.f64}\left(\left(e \cdot e\right), \color{blue}{\left(1 \cdot 1 - e \cdot 1\right)}\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(e, \mathsf{*.f64}\left(e, e\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(e, e\right), \left(\color{blue}{1 \cdot 1} - e \cdot 1\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(e, \mathsf{*.f64}\left(e, e\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(e, e\right), \left(1 - \color{blue}{e} \cdot 1\right)\right)\right) \]

   16. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(e, \mathsf{*.f64}\left(e, e\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(e, e\right), \left(1 - e\right)\right)\right) \]

   17. --lowering--.f64 99.1%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(e, \mathsf{*.f64}\left(e, e\right)\right)\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(e, e\right), \mathsf{\_.f64}\left(1, \color{blue}{e}\right)\right)\right) \]

7. Applied egg-rr 99.1%

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

Alternative 6: 98.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in v around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \color{blue}{\left(1 + e\right)}\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \left(e + \color{blue}{1}\right)\right) \]

   2. +-lowering-+.f64 99.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right), \mathsf{+.f64}\left(e, \color{blue}{1}\right)\right) \]

5. Simplified 99.0%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sin v}{e + 1}\right), \color{blue}{e}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin v, \left(e + 1\right)\right), e\right) \]

   5. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(v\right), \left(e + 1\right)\right), e\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(v\right), \left(1 + e\right)\right), e\right) \]

   7. +-lowering-+.f64 99.1%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(v\right), \mathsf{+.f64}\left(1, e\right)\right), e\right) \]

7. Applied egg-rr 99.1%

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

8. Final simplification 99.1%

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

Alternative 7: 97.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in e around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(e, \color{blue}{\sin v}\right) \]

   2. sin-lowering-sin.f64 97.8%

      \[\leadsto \mathsf{*.f64}\left(e, \mathsf{sin.f64}\left(v\right)\right) \]

5. Simplified 97.8%

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

Alternative 8: 52.4% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    code = e / (((e + 1.0d0) + ((v * v) * ((e * (-0.5d0)) + ((e + 1.0d0) * 0.16666666666666666d0)))) / v)
end function

Java

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

Python

def code(e, v):
	return e / (((e + 1.0) + ((v * v) * ((e * -0.5) + ((e + 1.0) * 0.16666666666666666)))) / v)

Julia

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

MATLAB

function tmp = code(e, v)
	tmp = e / (((e + 1.0) + ((v * v) * ((e * -0.5) + ((e + 1.0) * 0.16666666666666666)))) / v);
end

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-/l* N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(e, \color{blue}{\left(\frac{1 + e \cdot \cos v}{\sin v}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\left(1 + e \cdot \cos v\right), \color{blue}{\sin v}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(e \cdot \cos v\right)\right), \sin \color{blue}{v}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(e, \cos v\right)\right), \sin v\right)\right) \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(e, \mathsf{cos.f64}\left(v\right)\right)\right), \sin v\right)\right) \]

   9. sin-lowering-sin.f64 99.6%

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(e, \mathsf{cos.f64}\left(v\right)\right)\right), \mathsf{sin.f64}\left(v\right)\right)\right) \]

4. Applied egg-rr 99.6%

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

5. Taylor expanded in v around 0

   \[\leadsto \mathsf{/.f64}\left(e, \color{blue}{\left(\frac{1 + \left(e + {v}^{2} \cdot \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right)\right)}{v}\right)}\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\left(1 + \left(e + {v}^{2} \cdot \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right)\right)\right), \color{blue}{v}\right)\right) \]

   2. associate-+r+ N/A

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\left(\left(1 + e\right) + {v}^{2} \cdot \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right)\right), v\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(1 + e\right), \left({v}^{2} \cdot \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right)\right)\right), v\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, e\right), \left({v}^{2} \cdot \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right)\right)\right), v\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, e\right), \mathsf{*.f64}\left(\left({v}^{2}\right), \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right)\right)\right), v\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, e\right), \mathsf{*.f64}\left(\left(v \cdot v\right), \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right)\right)\right), v\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, e\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\frac{-1}{2} \cdot e - \frac{-1}{6} \cdot \left(1 + e\right)\right)\right)\right), v\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, e\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \left(\frac{-1}{2} \cdot e + \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \left(1 + e\right)\right)\right)\right)\right)\right), v\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, e\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot e\right), \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \left(1 + e\right)\right)\right)\right)\right)\right), v\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, e\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{+.f64}\left(\left(e \cdot \frac{-1}{2}\right), \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \left(1 + e\right)\right)\right)\right)\right)\right), v\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, e\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(e, \frac{-1}{2}\right), \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \left(1 + e\right)\right)\right)\right)\right)\right), v\right)\right) \]

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

       \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, e\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(e, \frac{-1}{2}\right), \left(\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot \left(1 + e\right)\right)\right)\right)\right), v\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, e\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(e, \frac{-1}{2}\right), \left(\frac{1}{6} \cdot \left(1 + e\right)\right)\right)\right)\right), v\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, e\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(e, \frac{-1}{2}\right), \mathsf{*.f64}\left(\frac{1}{6}, \left(1 + e\right)\right)\right)\right)\right), v\right)\right) \]

   15. +-lowering-+.f64 49.0%

       \[\leadsto \mathsf{/.f64}\left(e, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, e\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(v, v\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(e, \frac{-1}{2}\right), \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{+.f64}\left(1, e\right)\right)\right)\right)\right), v\right)\right) \]

7. Simplified 49.0%

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

8. Final simplification 49.0%

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

Alternative 9: 51.3% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

      \[\leadsto \mathsf{/.f64}\left(\left(e \cdot v\right), \color{blue}{\left(1 + e\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, v\right), \left(\color{blue}{1} + e\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, v\right), \left(e + \color{blue}{1}\right)\right) \]

   4. +-lowering-+.f64 47.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, v\right), \mathsf{+.f64}\left(e, \color{blue}{1}\right)\right) \]

5. Simplified 47.9%

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

   1. associate-*l/ N/A

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

   2. *-commutative N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(v, \color{blue}{\left(\frac{e + 1}{e}\right)}\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(v, \mathsf{/.f64}\left(\left(e + 1\right), \color{blue}{e}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(v, \mathsf{/.f64}\left(\left(1 + e\right), e\right)\right) \]

   8. +-lowering-+.f64 47.9%

      \[\leadsto \mathsf{/.f64}\left(v, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, e\right), e\right)\right) \]

7. Applied egg-rr 47.9%

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

   1. associate-/r/ N/A

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

   2. associate-*l/ N/A

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

   3. *-commutative N/A

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

   4. +-commutative N/A

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

   5. flip3-+ N/A

      \[\leadsto \frac{e \cdot v}{\frac{{e}^{3} + {1}^{3}}{\color{blue}{e \cdot e + \left(1 \cdot 1 - e \cdot 1\right)}}} \]

   6. div-inv N/A

      \[\leadsto \frac{e \cdot v}{\left({e}^{3} + {1}^{3}\right) \cdot \color{blue}{\frac{1}{e \cdot e + \left(1 \cdot 1 - e \cdot 1\right)}}} \]

   7. times-frac N/A

      \[\leadsto \frac{e}{{e}^{3} + {1}^{3}} \cdot \color{blue}{\frac{v}{\frac{1}{e \cdot e + \left(1 \cdot 1 - e \cdot 1\right)}}} \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{e}{{e}^{3} + {1}^{3}}\right), \color{blue}{\left(\frac{v}{\frac{1}{e \cdot e + \left(1 \cdot 1 - e \cdot 1\right)}}\right)}\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(e, \left({e}^{3} + {1}^{3}\right)\right), \left(\frac{\color{blue}{v}}{\frac{1}{e \cdot e + \left(1 \cdot 1 - e \cdot 1\right)}}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(e, \left({e}^{3} + 1\right)\right), \left(\frac{v}{\frac{1}{e \cdot e + \left(1 \cdot 1 - e \cdot 1\right)}}\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(e, \left(1 + {e}^{3}\right)\right), \left(\frac{v}{\frac{1}{e \cdot e + \left(1 \cdot 1 - e \cdot 1\right)}}\right)\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(e, \mathsf{+.f64}\left(1, \left({e}^{3}\right)\right)\right), \left(\frac{v}{\frac{1}{e \cdot e + \left(1 \cdot 1 - e \cdot 1\right)}}\right)\right) \]

   13. cube-mult N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(e, \mathsf{+.f64}\left(1, \left(e \cdot \left(e \cdot e\right)\right)\right)\right), \left(\frac{v}{\frac{1}{e \cdot e + \left(1 \cdot 1 - e \cdot 1\right)}}\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(e, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(e, \left(e \cdot e\right)\right)\right)\right), \left(\frac{v}{\frac{1}{e \cdot e + \left(1 \cdot 1 - e \cdot 1\right)}}\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(e, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(e, \mathsf{*.f64}\left(e, e\right)\right)\right)\right), \left(\frac{v}{\frac{1}{e \cdot e + \left(1 \cdot 1 - e \cdot 1\right)}}\right)\right) \]

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(e, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(e, \mathsf{*.f64}\left(e, e\right)\right)\right)\right), \mathsf{/.f64}\left(v, \color{blue}{\left(\frac{1}{e \cdot e + \left(1 \cdot 1 - e \cdot 1\right)}\right)}\right)\right) \]

   17. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(e, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(e, \mathsf{*.f64}\left(e, e\right)\right)\right)\right), \mathsf{/.f64}\left(v, \mathsf{/.f64}\left(1, \color{blue}{\left(e \cdot e + \left(1 \cdot 1 - e \cdot 1\right)\right)}\right)\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(e, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(e, \mathsf{*.f64}\left(e, e\right)\right)\right)\right), \mathsf{/.f64}\left(v, \mathsf{/.f64}\left(1, \left(e \cdot e + \left(1 - \color{blue}{e} \cdot 1\right)\right)\right)\right)\right) \]

   19. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(e, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(e, \mathsf{*.f64}\left(e, e\right)\right)\right)\right), \mathsf{/.f64}\left(v, \mathsf{/.f64}\left(1, \left(e \cdot e + \left(1 - e\right)\right)\right)\right)\right) \]

   20. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(e, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(e, \mathsf{*.f64}\left(e, e\right)\right)\right)\right), \mathsf{/.f64}\left(v, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(e \cdot e\right), \color{blue}{\left(1 - e\right)}\right)\right)\right)\right) \]

   21. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(e, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(e, \mathsf{*.f64}\left(e, e\right)\right)\right)\right), \mathsf{/.f64}\left(v, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(e, e\right), \left(\color{blue}{1} - e\right)\right)\right)\right)\right) \]

9. Applied egg-rr 48.0%

   \[\leadsto \color{blue}{\frac{e}{1 + e \cdot \left(e \cdot e\right)} \cdot \frac{v}{\frac{1}{e \cdot e + \left(1 - e\right)}}} \]
10. Add Preprocessing

Alternative 10: 51.3% accurate, 29.9× 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 99.7%

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

      \[\leadsto \mathsf{/.f64}\left(\left(e \cdot v\right), \color{blue}{\left(1 + e\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, v\right), \left(\color{blue}{1} + e\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, v\right), \left(e + \color{blue}{1}\right)\right) \]

   4. +-lowering-+.f64 47.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, v\right), \mathsf{+.f64}\left(e, \color{blue}{1}\right)\right) \]

5. Simplified 47.9%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{v}{e + 1}\right), \color{blue}{e}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(v, \left(e + 1\right)\right), e\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(v, \left(1 + e\right)\right), e\right) \]

   6. +-lowering-+.f64 47.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(v, \mathsf{+.f64}\left(1, e\right)\right), e\right) \]

7. Applied egg-rr 47.9%

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

8. Final simplification 47.9%

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

Alternative 11: 50.7% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

      \[\leadsto \mathsf{/.f64}\left(\left(e \cdot v\right), \color{blue}{\left(1 + e\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, v\right), \left(\color{blue}{1} + e\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, v\right), \left(e + \color{blue}{1}\right)\right) \]

   4. +-lowering-+.f64 47.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, v\right), \mathsf{+.f64}\left(e, \color{blue}{1}\right)\right) \]

5. Simplified 47.9%

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

6. Taylor expanded in e around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(e, \color{blue}{\left(v + -1 \cdot \left(e \cdot v\right)\right)}\right) \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(e, \left(v + \left(\mathsf{neg}\left(e \cdot v\right)\right)\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(e, \left(v - \color{blue}{e \cdot v}\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(e, \mathsf{\_.f64}\left(v, \color{blue}{\left(e \cdot v\right)}\right)\right) \]

   5. *-lowering-*.f64 47.4%

      \[\leadsto \mathsf{*.f64}\left(e, \mathsf{\_.f64}\left(v, \mathsf{*.f64}\left(e, \color{blue}{v}\right)\right)\right) \]

8. Simplified 47.4%

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

Alternative 12: 50.2% accurate, 69.7× 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 99.7%

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

      \[\leadsto \mathsf{/.f64}\left(\left(e \cdot v\right), \color{blue}{\left(1 + e\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, v\right), \left(\color{blue}{1} + e\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, v\right), \left(e + \color{blue}{1}\right)\right) \]

   4. +-lowering-+.f64 47.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, v\right), \mathsf{+.f64}\left(e, \color{blue}{1}\right)\right) \]

5. Simplified 47.9%

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

6. Taylor expanded in e around 0

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

   1. *-lowering-*.f64 46.6%

      \[\leadsto \mathsf{*.f64}\left(e, \color{blue}{v}\right) \]

8. Simplified 46.6%

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

Alternative 13: 4.5% accurate, 209.0× speedup

Math

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

FPCore

(FPCore (e v) :precision binary64 v)

C

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

Fortran

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

Java

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

Python

def code(e, v):
	return v

Julia

function code(e, v)
	return v
end

MATLAB

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

Wolfram

code[e_, v_] := v

Derivation

1. Initial program 99.7%

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

      \[\leadsto \mathsf{/.f64}\left(\left(e \cdot v\right), \color{blue}{\left(1 + e\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, v\right), \left(\color{blue}{1} + e\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, v\right), \left(e + \color{blue}{1}\right)\right) \]

   4. +-lowering-+.f64 47.9%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(e, v\right), \mathsf{+.f64}\left(e, \color{blue}{1}\right)\right) \]

5. Simplified 47.9%

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

6. Taylor expanded in e around inf

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

8. Simplified 4.3%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))