2sin (example 3.3)

Percentage Accurate: 42.1% → 99.4%
Time: 11.7s
Alternatives: 9
Speedup: 2.0×

Specification

?
\[\begin{array}{l} \\ \sin \left(x + \varepsilon\right) - \sin x \end{array} \]
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
double code(double x, double eps) {
	return sin((x + eps)) - sin(x);
}

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = sin((x + eps)) - sin(x)
end function

public static double code(double x, double eps) {
	return Math.sin((x + eps)) - Math.sin(x);
}

def code(x, eps):
	return math.sin((x + eps)) - math.sin(x)

function code(x, eps)
	return Float64(sin(Float64(x + eps)) - sin(x))
end

function tmp = code(x, eps)
	tmp = sin((x + eps)) - sin(x);
end

code[x_, eps_] := N[(N[Sin[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\sin \left(x + \varepsilon\right) - \sin x
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 9 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 42.1% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \sin \left(x + \varepsilon\right) - \sin x \end{array} \]
(FPCore (x eps) :precision binary64 (- (sin (+ x eps)) (sin x)))
double code(double x, double eps) {
	return sin((x + eps)) - sin(x);
}

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = sin((x + eps)) - sin(x)
end function

public static double code(double x, double eps) {
	return Math.sin((x + eps)) - Math.sin(x);
}

def code(x, eps):
	return math.sin((x + eps)) - math.sin(x)

function code(x, eps)
	return Float64(sin(Float64(x + eps)) - sin(x))
end

function tmp = code(x, eps)
	tmp = sin((x + eps)) - sin(x);
end

code[x_, eps_] := N[(N[Sin[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\sin \left(x + \varepsilon\right) - \sin x
\end{array}

Alternative 1: 99.4% accurate, 0.3× speedup?

\[\begin{array}{l} \\ \sin \varepsilon \cdot \cos x - \sin x \cdot e^{\mathsf{log1p}\left(-\cos \varepsilon\right)} \end{array} \]
(FPCore (x eps)
 :precision binary64
 (- (* (sin eps) (cos x)) (* (sin x) (exp (log1p (- (cos eps)))))))
double code(double x, double eps) {
	return (sin(eps) * cos(x)) - (sin(x) * exp(log1p(-cos(eps))));
}

public static double code(double x, double eps) {
	return (Math.sin(eps) * Math.cos(x)) - (Math.sin(x) * Math.exp(Math.log1p(-Math.cos(eps))));
}

def code(x, eps):
	return (math.sin(eps) * math.cos(x)) - (math.sin(x) * math.exp(math.log1p(-math.cos(eps))))

function code(x, eps)
	return Float64(Float64(sin(eps) * cos(x)) - Float64(sin(x) * exp(log1p(Float64(-cos(eps))))))
end

code[x_, eps_] := N[(N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * N[Exp[N[Log[1 + (-N[Cos[eps], $MachinePrecision])], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\sin \varepsilon \cdot \cos x - \sin x \cdot e^{\mathsf{log1p}\left(-\cos \varepsilon\right)}
\end{array}
Derivation
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  1. Add Preprocessing

Alternative 2: 76.6% accurate, 0.3× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{if}\;t_0 \leq -0.005 \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \end{array} \]
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin (+ eps x)) (sin x))))
   (if (or (<= t_0 -0.005) (not (<= t_0 0.0)))
     t_0
     (* (cos x) (* 2.0 (sin (* eps 0.5)))))))
double code(double x, double eps) {
	double t_0 = sin((eps + x)) - sin(x);
	double tmp;
	if ((t_0 <= -0.005) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	}
	return tmp;
}

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((eps + x)) - sin(x)
    if ((t_0 <= (-0.005d0)) .or. (.not. (t_0 <= 0.0d0))) then
        tmp = t_0
    else
        tmp = cos(x) * (2.0d0 * sin((eps * 0.5d0)))
    end if
    code = tmp
end function

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps + x)) - Math.sin(x);
	double tmp;
	if ((t_0 <= -0.005) || !(t_0 <= 0.0)) {
		tmp = t_0;
	} else {
		tmp = Math.cos(x) * (2.0 * Math.sin((eps * 0.5)));
	}
	return tmp;
}

def code(x, eps):
	t_0 = math.sin((eps + x)) - math.sin(x)
	tmp = 0
	if (t_0 <= -0.005) or not (t_0 <= 0.0):
		tmp = t_0
	else:
		tmp = math.cos(x) * (2.0 * math.sin((eps * 0.5)))
	return tmp

function code(x, eps)
	t_0 = Float64(sin(Float64(eps + x)) - sin(x))
	tmp = 0.0
	if ((t_0 <= -0.005) || !(t_0 <= 0.0))
		tmp = t_0;
	else
		tmp = Float64(cos(x) * Float64(2.0 * sin(Float64(eps * 0.5))));
	end
	return tmp
end

function tmp_2 = code(x, eps)
	t_0 = sin((eps + x)) - sin(x);
	tmp = 0.0;
	if ((t_0 <= -0.005) || ~((t_0 <= 0.0)))
		tmp = t_0;
	else
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	end
	tmp_2 = tmp;
end

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.005], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], t$95$0, N[(N[Cos[x], $MachinePrecision] * N[(2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \sin \left(\varepsilon + x\right) - \sin x\\
\mathbf{if}\;t_0 \leq -0.005 \lor \neg \left(t_0 \leq 0\right):\\
\;\;\;\;t_0\\

\mathbf{else}:\\
\;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\


\end{array}
\end{array}
Derivation
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  1. Add Preprocessing

Alternative 3: 99.4% accurate, 0.4× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(\sin x, \cos \varepsilon + -1, \sin \varepsilon \cdot \cos x\right) \end{array} \]
(FPCore (x eps)
 :precision binary64
 (fma (sin x) (+ (cos eps) -1.0) (* (sin eps) (cos x))))
double code(double x, double eps) {
	return fma(sin(x), (cos(eps) + -1.0), (sin(eps) * cos(x)));
}

function code(x, eps)
	return fma(sin(x), Float64(cos(eps) + -1.0), Float64(sin(eps) * cos(x)))
end

code[x_, eps_] := N[(N[Sin[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\mathsf{fma}\left(\sin x, \cos \varepsilon + -1, \sin \varepsilon \cdot \cos x\right)
\end{array}
Derivation
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Alternative 4: 99.4% accurate, 0.5× speedup?

\[\begin{array}{l} \\ \sin \varepsilon \cdot \cos x - \sin x \cdot \left(1 - \cos \varepsilon\right) \end{array} \]
(FPCore (x eps)
 :precision binary64
 (- (* (sin eps) (cos x)) (* (sin x) (- 1.0 (cos eps)))))
double code(double x, double eps) {
	return (sin(eps) * cos(x)) - (sin(x) * (1.0 - cos(eps)));
}

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (sin(eps) * cos(x)) - (sin(x) * (1.0d0 - cos(eps)))
end function

public static double code(double x, double eps) {
	return (Math.sin(eps) * Math.cos(x)) - (Math.sin(x) * (1.0 - Math.cos(eps)));
}

def code(x, eps):
	return (math.sin(eps) * math.cos(x)) - (math.sin(x) * (1.0 - math.cos(eps)))

function code(x, eps)
	return Float64(Float64(sin(eps) * cos(x)) - Float64(sin(x) * Float64(1.0 - cos(eps))))
end

function tmp = code(x, eps)
	tmp = (sin(eps) * cos(x)) - (sin(x) * (1.0 - cos(eps)));
end

code[x_, eps_] := N[(N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * N[(1.0 - N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\sin \varepsilon \cdot \cos x - \sin x \cdot \left(1 - \cos \varepsilon\right)
\end{array}
Derivation
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Alternative 5: 77.6% accurate, 0.5× speedup?

\[\begin{array}{l} \\ \sin \varepsilon \cdot \cos x + \left(\sin x - \sin x\right) \end{array} \]
(FPCore (x eps)
 :precision binary64
 (+ (* (sin eps) (cos x)) (- (sin x) (sin x))))
double code(double x, double eps) {
	return (sin(eps) * cos(x)) + (sin(x) - sin(x));
}

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (sin(eps) * cos(x)) + (sin(x) - sin(x))
end function

public static double code(double x, double eps) {
	return (Math.sin(eps) * Math.cos(x)) + (Math.sin(x) - Math.sin(x));
}

def code(x, eps):
	return (math.sin(eps) * math.cos(x)) + (math.sin(x) - math.sin(x))

function code(x, eps)
	return Float64(Float64(sin(eps) * cos(x)) + Float64(sin(x) - sin(x)))
end

function tmp = code(x, eps)
	tmp = (sin(eps) * cos(x)) + (sin(x) - sin(x));
end

code[x_, eps_] := N[(N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\sin \varepsilon \cdot \cos x + \left(\sin x - \sin x\right)
\end{array}
Derivation
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Alternative 6: 76.4% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \cos \left(x + \varepsilon \cdot 0.5\right) \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \end{array} \]
(FPCore (x eps)
 :precision binary64
 (* (cos (+ x (* eps 0.5))) (* 2.0 (sin (* eps 0.5)))))
double code(double x, double eps) {
	return cos((x + (eps * 0.5))) * (2.0 * sin((eps * 0.5)));
}

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = cos((x + (eps * 0.5d0))) * (2.0d0 * sin((eps * 0.5d0)))
end function

public static double code(double x, double eps) {
	return Math.cos((x + (eps * 0.5))) * (2.0 * Math.sin((eps * 0.5)));
}

def code(x, eps):
	return math.cos((x + (eps * 0.5))) * (2.0 * math.sin((eps * 0.5)))

function code(x, eps)
	return Float64(cos(Float64(x + Float64(eps * 0.5))) * Float64(2.0 * sin(Float64(eps * 0.5))))
end

function tmp = code(x, eps)
	tmp = cos((x + (eps * 0.5))) * (2.0 * sin((eps * 0.5)));
end

code[x_, eps_] := N[(N[Cos[N[(x + N[(eps * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\cos \left(x + \varepsilon \cdot 0.5\right) \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)
\end{array}
Derivation
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Alternative 7: 76.3% accurate, 1.9× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00013 \lor \neg \left(\varepsilon \leq 2.3 \cdot 10^{-11}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]
(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00013) (not (<= eps 2.3e-11))) (sin eps) (* eps (cos x))))
double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00013) || !(eps <= 2.3e-11)) {
		tmp = sin(eps);
	} else {
		tmp = eps * cos(x);
	}
	return tmp;
}

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.00013d0)) .or. (.not. (eps <= 2.3d-11))) then
        tmp = sin(eps)
    else
        tmp = eps * cos(x)
    end if
    code = tmp
end function

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00013) || !(eps <= 2.3e-11)) {
		tmp = Math.sin(eps);
	} else {
		tmp = eps * Math.cos(x);
	}
	return tmp;
}

def code(x, eps):
	tmp = 0
	if (eps <= -0.00013) or not (eps <= 2.3e-11):
		tmp = math.sin(eps)
	else:
		tmp = eps * math.cos(x)
	return tmp

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00013) || !(eps <= 2.3e-11))
		tmp = sin(eps);
	else
		tmp = Float64(eps * cos(x));
	end
	return tmp
end

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.00013) || ~((eps <= 2.3e-11)))
		tmp = sin(eps);
	else
		tmp = eps * cos(x);
	end
	tmp_2 = tmp;
end

code[x_, eps_] := If[Or[LessEqual[eps, -0.00013], N[Not[LessEqual[eps, 2.3e-11]], $MachinePrecision]], N[Sin[eps], $MachinePrecision], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\varepsilon \leq -0.00013 \lor \neg \left(\varepsilon \leq 2.3 \cdot 10^{-11}\right):\\
\;\;\;\;\sin \varepsilon\\

\mathbf{else}:\\
\;\;\;\;\varepsilon \cdot \cos x\\


\end{array}
\end{array}
Derivation
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Alternative 8: 55.2% accurate, 2.0× speedup?

\[\begin{array}{l} \\ \sin \varepsilon \end{array} \]
(FPCore (x eps) :precision binary64 (sin eps))
double code(double x, double eps) {
	return sin(eps);
}

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = sin(eps)
end function

public static double code(double x, double eps) {
	return Math.sin(eps);
}

def code(x, eps):
	return math.sin(eps)

function code(x, eps)
	return sin(eps)
end

function tmp = code(x, eps)
	tmp = sin(eps);
end

code[x_, eps_] := N[Sin[eps], $MachinePrecision]

\begin{array}{l}

\\
\sin \varepsilon
\end{array}
Derivation
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Alternative 9: 29.7% accurate, 205.0× speedup?

\[\begin{array}{l} \\ \varepsilon \end{array} \]
(FPCore (x eps) :precision binary64 eps)
double code(double x, double eps) {
	return eps;
}

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps
end function

public static double code(double x, double eps) {
	return eps;
}

def code(x, eps):
	return eps

function code(x, eps)
	return eps
end

function tmp = code(x, eps)
	tmp = eps;
end

code[x_, eps_] := eps

\begin{array}{l}

\\
\varepsilon
\end{array}
Derivation
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Developer target: 99.4% accurate, 0.4× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \sin \varepsilon \cdot \cos x\right) \end{array} \]
(FPCore (x eps)
 :precision binary64
 (fma (sin x) (- (cos eps) 1.0) (* (sin eps) (cos x))))
double code(double x, double eps) {
	return fma(sin(x), (cos(eps) - 1.0), (sin(eps) * cos(x)));
}

function code(x, eps)
	return fma(sin(x), Float64(cos(eps) - 1.0), Float64(sin(eps) * cos(x)))
end

code[x_, eps_] := N[(N[Sin[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] - 1.0), $MachinePrecision] + N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\mathsf{fma}\left(\sin x, \cos \varepsilon - 1, \sin \varepsilon \cdot \cos x\right)
\end{array}

Reproduce

?
herbie shell --seed 2023342 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (fma (sin x) (- (cos eps) 1.0) (* (sin eps) (cos x)))

  (- (sin (+ x eps)) (sin x)))