sintan (problem 3.4.5)

Percentage Accurate: 1.6% → 99.7%
Time: 29.2s
Alternatives: 2
Speedup: 207.0×

Specification

?
\[-0.4 \leq \varepsilon \land \varepsilon \leq 0.4\]
\[\begin{array}{l} \\ \frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \end{array} \]
(FPCore (eps) :precision binary64 (/ (- eps (sin eps)) (- eps (tan eps))))
double code(double eps) {
	return (eps - sin(eps)) / (eps - tan(eps));
}

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = (eps - sin(eps)) / (eps - tan(eps))
end function

public static double code(double eps) {
	return (eps - Math.sin(eps)) / (eps - Math.tan(eps));
}

def code(eps):
	return (eps - math.sin(eps)) / (eps - math.tan(eps))

function code(eps)
	return Float64(Float64(eps - sin(eps)) / Float64(eps - tan(eps)))
end

function tmp = code(eps)
	tmp = (eps - sin(eps)) / (eps - tan(eps));
end

code[eps_] := N[(N[(eps - N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[(eps - N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon}
\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 2 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: 1.6% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \end{array} \]
(FPCore (eps) :precision binary64 (/ (- eps (sin eps)) (- eps (tan eps))))
double code(double eps) {
	return (eps - sin(eps)) / (eps - tan(eps));
}

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = (eps - sin(eps)) / (eps - tan(eps))
end function

public static double code(double eps) {
	return (eps - Math.sin(eps)) / (eps - Math.tan(eps));
}

def code(eps):
	return (eps - math.sin(eps)) / (eps - math.tan(eps))

function code(eps)
	return Float64(Float64(eps - sin(eps)) / Float64(eps - tan(eps)))
end

function tmp = code(eps)
	tmp = (eps - sin(eps)) / (eps - tan(eps));
end

code[eps_] := N[(N[(eps - N[Sin[eps], $MachinePrecision]), $MachinePrecision] / N[(eps - N[Tan[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon}
\end{array}

Alternative 1: 99.7% accurate, 2.0× speedup?

\[\begin{array}{l} \\ 0.225 \cdot {\varepsilon}^{2} - 0.5 \end{array} \]
(FPCore (eps) :precision binary64 (- (* 0.225 (pow eps 2.0)) 0.5))
double code(double eps) {
	return (0.225 * pow(eps, 2.0)) - 0.5;
}

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = (0.225d0 * (eps ** 2.0d0)) - 0.5d0
end function

public static double code(double eps) {
	return (0.225 * Math.pow(eps, 2.0)) - 0.5;
}

def code(eps):
	return (0.225 * math.pow(eps, 2.0)) - 0.5

function code(eps)
	return Float64(Float64(0.225 * (eps ^ 2.0)) - 0.5)
end

function tmp = code(eps)
	tmp = (0.225 * (eps ^ 2.0)) - 0.5;
end

code[eps_] := N[(N[(0.225 * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]

\begin{array}{l}

\\
0.225 \cdot {\varepsilon}^{2} - 0.5
\end{array}
Derivation

  1. Initial program 2.0%

    \[\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \]
  2. Add Preprocessing

  3. Taylor expanded in eps around 0 99.5%

    \[\leadsto \color{blue}{0.225 \cdot {\varepsilon}^{2} - 0.5} \]

  4. Final simplification99.5%

    \[\leadsto 0.225 \cdot {\varepsilon}^{2} - 0.5 \]
  5. Add Preprocessing

Alternative 2: 99.1% accurate, 207.0× speedup?

\[\begin{array}{l} \\ -0.5 \end{array} \]
(FPCore (eps) :precision binary64 -0.5)
double code(double eps) {
	return -0.5;
}

real(8) function code(eps)
    real(8), intent (in) :: eps
    code = -0.5d0
end function

public static double code(double eps) {
	return -0.5;
}

def code(eps):
	return -0.5

function code(eps)
	return -0.5
end

function tmp = code(eps)
	tmp = -0.5;
end

code[eps_] := -0.5

\begin{array}{l}

\\
-0.5
\end{array}
Derivation

  1. Initial program 2.0%

    \[\frac{\varepsilon - \sin \varepsilon}{\varepsilon - \tan \varepsilon} \]
  2. Add Preprocessing

  3. Taylor expanded in eps around 0 99.2%

    \[\leadsto \color{blue}{-0.5} \]

  4. Final simplification99.2%

    \[\leadsto -0.5 \]
  5. Add Preprocessing

Developer target: 99.9% accurate, 5.6× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \left(\left(-0.5 + \frac{9 \cdot \left(\varepsilon \cdot \varepsilon\right)}{40}\right) + \frac{-27 \cdot t\_0}{2800}\right) + \frac{27 \cdot \left(\left(t\_0 \cdot \varepsilon\right) \cdot \varepsilon\right)}{112000} \end{array} \end{array} \]
(FPCore (eps)
 :precision binary64
 (let* ((t_0 (* (* (* eps eps) eps) eps)))
   (+
    (+ (+ -0.5 (/ (* 9.0 (* eps eps)) 40.0)) (/ (* -27.0 t_0) 2800.0))
    (/ (* 27.0 (* (* t_0 eps) eps)) 112000.0))))
double code(double eps) {
	double t_0 = ((eps * eps) * eps) * eps;
	return ((-0.5 + ((9.0 * (eps * eps)) / 40.0)) + ((-27.0 * t_0) / 2800.0)) + ((27.0 * ((t_0 * eps) * eps)) / 112000.0);
}

real(8) function code(eps)
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = ((eps * eps) * eps) * eps
    code = (((-0.5d0) + ((9.0d0 * (eps * eps)) / 40.0d0)) + (((-27.0d0) * t_0) / 2800.0d0)) + ((27.0d0 * ((t_0 * eps) * eps)) / 112000.0d0)
end function

public static double code(double eps) {
	double t_0 = ((eps * eps) * eps) * eps;
	return ((-0.5 + ((9.0 * (eps * eps)) / 40.0)) + ((-27.0 * t_0) / 2800.0)) + ((27.0 * ((t_0 * eps) * eps)) / 112000.0);
}

def code(eps):
	t_0 = ((eps * eps) * eps) * eps
	return ((-0.5 + ((9.0 * (eps * eps)) / 40.0)) + ((-27.0 * t_0) / 2800.0)) + ((27.0 * ((t_0 * eps) * eps)) / 112000.0)

function code(eps)
	t_0 = Float64(Float64(Float64(eps * eps) * eps) * eps)
	return Float64(Float64(Float64(-0.5 + Float64(Float64(9.0 * Float64(eps * eps)) / 40.0)) + Float64(Float64(-27.0 * t_0) / 2800.0)) + Float64(Float64(27.0 * Float64(Float64(t_0 * eps) * eps)) / 112000.0))
end

function tmp = code(eps)
	t_0 = ((eps * eps) * eps) * eps;
	tmp = ((-0.5 + ((9.0 * (eps * eps)) / 40.0)) + ((-27.0 * t_0) / 2800.0)) + ((27.0 * ((t_0 * eps) * eps)) / 112000.0);
end

code[eps_] := Block[{t$95$0 = N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision]}, N[(N[(N[(-0.5 + N[(N[(9.0 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] / 40.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-27.0 * t$95$0), $MachinePrecision] / 2800.0), $MachinePrecision]), $MachinePrecision] + N[(N[(27.0 * N[(N[(t$95$0 * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] / 112000.0), $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\
\left(\left(-0.5 + \frac{9 \cdot \left(\varepsilon \cdot \varepsilon\right)}{40}\right) + \frac{-27 \cdot t\_0}{2800}\right) + \frac{27 \cdot \left(\left(t\_0 \cdot \varepsilon\right) \cdot \varepsilon\right)}{112000}
\end{array}
\end{array}

Reproduce

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herbie shell --seed 2024076 
(FPCore (eps)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  :pre (and (<= -0.4 eps) (<= eps 0.4))

  :alt
  (+ (+ (+ -0.5 (/ (* 9.0 (* eps eps)) 40.0)) (/ (* -27.0 (* (* (* eps eps) eps) eps)) 2800.0)) (/ (* 27.0 (* (* (* (* (* eps eps) eps) eps) eps) eps)) 112000.0))

  (/ (- eps (sin eps)) (- eps (tan eps))))