Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 7.1s

Alternatives: 11

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-/l* 99.7%

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

   2. remove-double-neg 99.7%

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

   3. cos-neg 99.7%

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

   4. distribute-frac-neg 99.7%

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

   5. sin-neg 99.7%

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

   6. distribute-neg-frac 99.7%

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

   7. sin-neg 99.7%

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

   8. remove-double-neg 99.7%

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

   9. +-commutative 99.7%

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

   10. cos-neg 99.7%

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

   11. fma-define 99.7%

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

3. Simplified 99.7%

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

Alternative 2: 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]

Derivation

1. Initial program 99.7%

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

Alternative 3: 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. Step-by-step derivation

   1. associate-/l* 99.7%

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

   2. remove-double-neg 99.7%

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

   3. cos-neg 99.7%

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

   4. distribute-frac-neg 99.7%

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

   5. sin-neg 99.7%

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

   6. distribute-neg-frac 99.7%

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

   7. sin-neg 99.7%

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

   8. remove-double-neg 99.7%

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

   9. +-commutative 99.7%

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

   10. cos-neg 99.7%

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

   11. fma-define 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in e around inf 99.6%

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

6. Taylor expanded in v around inf 99.6%

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

Alternative 4: 74.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;v \leq 4 \cdot 10^{-28}:\\ \;\;\;\;v \cdot \frac{e}{e + 1}\\ \mathbf{else}:\\ \;\;\;\;e \cdot \sin v\\ \end{array} \end{array} \]

FPCore

(FPCore (e v)
 :precision binary64
 (if (<= v 4e-28) (* v (/ e (+ e 1.0))) (* e (sin v))))

C

double code(double e, double v) {
	double tmp;
	if (v <= 4e-28) {
		tmp = v * (e / (e + 1.0));
	} else {
		tmp = e * sin(v);
	}
	return tmp;
}

Fortran

real(8) function code(e, v)
    real(8), intent (in) :: e
    real(8), intent (in) :: v
    real(8) :: tmp
    if (v <= 4d-28) then
        tmp = v * (e / (e + 1.0d0))
    else
        tmp = e * sin(v)
    end if
    code = tmp
end function

Java

public static double code(double e, double v) {
	double tmp;
	if (v <= 4e-28) {
		tmp = v * (e / (e + 1.0));
	} else {
		tmp = e * Math.sin(v);
	}
	return tmp;
}

Python

def code(e, v):
	tmp = 0
	if v <= 4e-28:
		tmp = v * (e / (e + 1.0))
	else:
		tmp = e * math.sin(v)
	return tmp

Julia

function code(e, v)
	tmp = 0.0
	if (v <= 4e-28)
		tmp = Float64(v * Float64(e / Float64(e + 1.0)));
	else
		tmp = Float64(e * sin(v));
	end
	return tmp
end

MATLAB

function tmp_2 = code(e, v)
	tmp = 0.0;
	if (v <= 4e-28)
		tmp = v * (e / (e + 1.0));
	else
		tmp = e * sin(v);
	end
	tmp_2 = tmp;
end

Wolfram

code[e_, v_] := If[LessEqual[v, 4e-28], N[(v * N[(e / N[(e + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(e * N[Sin[v], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < 3.99999999999999988e-28

   1. Initial program 99.8%

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

      1. associate-/l* 99.8%

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

      2. remove-double-neg 99.8%

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

      3. cos-neg 99.8%

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

      4. distribute-frac-neg 99.8%

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

      5. sin-neg 99.8%

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

      6. distribute-neg-frac 99.8%

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

      7. sin-neg 99.8%

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

      8. remove-double-neg 99.8%

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

      9. +-commutative 99.8%

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

      10. cos-neg 99.8%

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

      11. fma-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around 0 70.7%

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

      1. *-commutative 70.7%

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

      2. associate-*r/ 70.7%

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

      3. +-commutative 70.7%

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

   7. Simplified 70.7%

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

   if 3.99999999999999988e-28 < v

   1. Initial program 99.6%

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

      1. associate-/l* 99.6%

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

      2. remove-double-neg 99.6%

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

      3. cos-neg 99.6%

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

      4. distribute-frac-neg 99.6%

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

      5. sin-neg 99.6%

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

      6. distribute-neg-frac 99.6%

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

      7. sin-neg 99.6%

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

      8. remove-double-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. cos-neg 99.6%

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

      11. fma-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in e around 0 99.6%

      \[\leadsto \color{blue}{e \cdot \sin v} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.7% accurate, 2.0× 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 99.7%

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

3. Taylor expanded in v around 0 99.4%

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

4. Final simplification 99.4%

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

Alternative 6: 98.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-/l* 99.7%

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

   2. remove-double-neg 99.7%

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

   3. cos-neg 99.7%

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

   4. distribute-frac-neg 99.7%

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

   5. sin-neg 99.7%

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

   6. distribute-neg-frac 99.7%

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

   7. sin-neg 99.7%

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

   8. remove-double-neg 99.7%

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

   9. +-commutative 99.7%

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

   10. cos-neg 99.7%

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

   11. fma-define 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in e around inf 99.6%

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

6. Taylor expanded in v around inf 99.6%

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

7. Taylor expanded in v around 0 99.2%

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

Alternative 7: 98.3% 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 99.7%

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

   1. associate-/l* 99.7%

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

   2. remove-double-neg 99.7%

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

   3. cos-neg 99.7%

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

   4. distribute-frac-neg 99.7%

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

   5. sin-neg 99.7%

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

   6. distribute-neg-frac 99.7%

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

   7. sin-neg 99.7%

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

   8. remove-double-neg 99.7%

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

   9. +-commutative 99.7%

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

   10. cos-neg 99.7%

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

   11. fma-define 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in e around 0 98.6%

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

   1. *-lft-identity 98.6%

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

   2. mul-1-neg 98.6%

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

   3. associate-*r* 98.6%

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

   4. distribute-lft-neg-in 98.6%

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

   5. distribute-rgt-in 98.6%

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

   6. sub-neg 98.6%

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

7. Simplified 98.6%

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

8. Taylor expanded in v around 0 98.3%

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

Alternative 8: 52.3% accurate, 29.9× 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 99.7%

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

   1. associate-/l* 99.7%

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

   2. remove-double-neg 99.7%

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

   3. cos-neg 99.7%

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

   4. distribute-frac-neg 99.7%

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

   5. sin-neg 99.7%

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

   6. distribute-neg-frac 99.7%

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

   7. sin-neg 99.7%

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

   8. remove-double-neg 99.7%

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

   9. +-commutative 99.7%

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

   10. cos-neg 99.7%

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

   11. fma-define 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in v around 0 55.7%

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

   1. *-commutative 55.7%

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

   2. associate-*r/ 55.8%

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

   3. +-commutative 55.8%

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

7. Simplified 55.8%

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

Alternative 9: 51.8% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-/l* 99.7%

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

   2. remove-double-neg 99.7%

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

   3. cos-neg 99.7%

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

   4. distribute-frac-neg 99.7%

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

   5. sin-neg 99.7%

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

   6. distribute-neg-frac 99.7%

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

   7. sin-neg 99.7%

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

   8. remove-double-neg 99.7%

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

   9. +-commutative 99.7%

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

   10. cos-neg 99.7%

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

   11. fma-define 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in v around 0 55.7%

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

   1. *-commutative 55.7%

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

   2. associate-*r/ 55.8%

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

   3. +-commutative 55.8%

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

7. Simplified 55.8%

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

8. Taylor expanded in e around 0 54.7%

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

   1. neg-mul-1 54.7%

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

   2. sub-neg 54.7%

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

10. Simplified 54.7%

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

Alternative 10: 51.3% 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. Step-by-step derivation

   1. associate-/l* 99.7%

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

   2. remove-double-neg 99.7%

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

   3. cos-neg 99.7%

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

   4. distribute-frac-neg 99.7%

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

   5. sin-neg 99.7%

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

   6. distribute-neg-frac 99.7%

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

   7. sin-neg 99.7%

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

   8. remove-double-neg 99.7%

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

   9. +-commutative 99.7%

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

   10. cos-neg 99.7%

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

   11. fma-define 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in v around 0 55.7%

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

   1. *-commutative 55.7%

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

   2. associate-*r/ 55.8%

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

   3. +-commutative 55.8%

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

7. Simplified 55.8%

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

8. Taylor expanded in e around 0 53.7%

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

9. Final simplification 53.7%

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

Alternative 11: 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. Step-by-step derivation

   1. associate-/l* 99.7%

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

   2. remove-double-neg 99.7%

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

   3. cos-neg 99.7%

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

   4. distribute-frac-neg 99.7%

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

   5. sin-neg 99.7%

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

   6. distribute-neg-frac 99.7%

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

   7. sin-neg 99.7%

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

   8. remove-double-neg 99.7%

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

   9. +-commutative 99.7%

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

   10. cos-neg 99.7%

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

   11. fma-define 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in v around 0 55.7%

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

   1. *-commutative 55.7%

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

   2. associate-*r/ 55.8%

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

   3. +-commutative 55.8%

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

7. Simplified 55.8%

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

   1. associate-*r/ 55.7%

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

   2. clear-num 54.8%

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

   3. *-commutative 54.8%

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

9. Applied egg-rr 54.8%

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

10. Taylor expanded in e around inf 4.8%

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

Reproduce

herbie shell --seed 2024182 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))