Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 9.3s

Alternatives: 9

Speedup: 1.0×

Specification

\[0 \leq e \land e \leq 1\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-cos.f64 99.8

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

5. Simplified 99.8%

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

Alternative 2: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in e around 0

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

   1. lower-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. associate-*r* N/A

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

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

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

   5. distribute-rgt1-in N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. mul-1-neg N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. mul-1-neg N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 98.9

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

5. Simplified 98.9%

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

6. Final simplification 98.9%

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

Alternative 3: 98.9% accurate, 1.9× speedup

Math

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

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(Float64(e * 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[(N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision] / N[(e + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around 0

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

5. Simplified 98.9%

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

6. Taylor expanded in v around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 98.9

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

8. Simplified 98.9%

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

Alternative 4: 98.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around 0

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

5. Simplified 98.9%

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

6. Taylor expanded in e around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-lft-identity N/A

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

   5. distribute-rgt-out N/A

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

   6. +-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mul-1-neg N/A

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

   10. unsub-neg N/A

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

   11. lower--.f64 98.4

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

8. Simplified 98.4%

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

Alternative 5: 97.9% accurate, 2.1× 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 99.8%

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

3. Taylor expanded in e around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 98.1

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

5. Simplified 98.1%

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

Alternative 6: 52.4% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 130000:\\ \;\;\;\;v \cdot \mathsf{fma}\left(v \cdot v, e \cdot -0.16666666666666666, \frac{e}{e + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;-e \cdot \left(e \cdot v\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (if (<= v 130000.0)
   (* v (fma (* v v) (* e -0.16666666666666666) (/ e (+ e 1.0))))
   (- (* e (* e v)))))

C

double code(double e, double v) {
	double tmp;
	if (v <= 130000.0) {
		tmp = v * fma((v * v), (e * -0.16666666666666666), (e / (e + 1.0)));
	} else {
		tmp = -(e * (e * v));
	}
	return tmp;
}

Julia

function code(e, v)
	tmp = 0.0
	if (v <= 130000.0)
		tmp = Float64(v * fma(Float64(v * v), Float64(e * -0.16666666666666666), Float64(e / Float64(e + 1.0))));
	else
		tmp = Float64(-Float64(e * Float64(e * v)));
	end
	return tmp
end

Wolfram

code[e_, v_] := If[LessEqual[v, 130000.0], N[(v * N[(N[(v * v), $MachinePrecision] * N[(e * -0.16666666666666666), $MachinePrecision] + N[(e / N[(e + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(e * N[(e * v), $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if v < 1.3e5

   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

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   5. Simplified 70.5%

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

   6. Taylor expanded in e around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 70.5

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

   8. Simplified 70.5%

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

   if 1.3e5 < v

   1. Initial program 99.6%

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

   3. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 3.5

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

   5. Simplified 3.5%

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

   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. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 3.5

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

   8. Simplified 3.5%

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

   9. Taylor expanded in e around inf

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

       1. mul-1-neg N/A

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

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

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

       3. mul-1-neg N/A

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

       4. lower-*.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f64 5.3

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

   11. Simplified 5.3%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 130000:\\ \;\;\;\;v \cdot \mathsf{fma}\left(v \cdot v, e \cdot -0.16666666666666666, \frac{e}{e + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;-e \cdot \left(e \cdot v\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.1% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 51.6

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

5. Simplified 51.6%

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

Alternative 8: 51.6% accurate, 16.1× 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 99.8%

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

3. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 51.6

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

5. Simplified 51.6%

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

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. lower-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 51.1

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

8. Simplified 51.1%

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

Alternative 9: 51.1% accurate, 37.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in e around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 98.1

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

5. Simplified 98.1%

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

6. Taylor expanded in v around 0

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

   1. lower-*.f64 50.8

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

8. Simplified 50.8%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))