Trigonometry A

Percentage Accurate: 99.8% → 99.8%

Time: 6.7s

Alternatives: 10

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

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

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

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1. Add Preprocessing

Alternative 2: 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

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Alternative 3: 97.7% 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

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Alternative 4: 52.1% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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Alternative 5: 52.1% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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Alternative 6: 50.8% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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Alternative 7: 50.8% accurate, 29.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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Alternative 8: 50.9% accurate, 29.9× speedup

Math

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

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(Float64(e * v) / Float64(e + 1.0))
end

MATLAB

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

Wolfram

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

Derivation

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Alternative 9: 49.8% 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

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1. Add Preprocessing

Alternative 10: 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

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1. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(FPCore (e v)
  :name "Trigonometry A"
  :precision binary64
  :pre (and (<= 0.0 e) (<= e 1.0))
  (/ (* e (sin v)) (+ 1.0 (* e (cos v)))))