
(FPCore (x) :precision binary64 (let* ((t_0 (sin (* x 0.5)))) (/ (* (* (/ 8.0 3.0) t_0) t_0) (sin x))))
double code(double x) {
double t_0 = sin((x * 0.5));
return (((8.0 / 3.0) * t_0) * t_0) / sin(x);
}
real(8) function code(x)
real(8), intent (in) :: x
real(8) :: t_0
t_0 = sin((x * 0.5d0))
code = (((8.0d0 / 3.0d0) * t_0) * t_0) / sin(x)
end function
public static double code(double x) {
double t_0 = Math.sin((x * 0.5));
return (((8.0 / 3.0) * t_0) * t_0) / Math.sin(x);
}
def code(x): t_0 = math.sin((x * 0.5)) return (((8.0 / 3.0) * t_0) * t_0) / math.sin(x)
function code(x) t_0 = sin(Float64(x * 0.5)) return Float64(Float64(Float64(Float64(8.0 / 3.0) * t_0) * t_0) / sin(x)) end
function tmp = code(x) t_0 = sin((x * 0.5)); tmp = (((8.0 / 3.0) * t_0) * t_0) / sin(x); end
code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(8.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
t_0 := \sin \left(x \cdot 0.5\right)\\
\frac{\left(\frac{8}{3} \cdot t_0\right) \cdot t_0}{\sin x}
\end{array}
\end{array}
Sampling outcomes in binary64 precision:
Herbie found 3 alternatives:
| Alternative | Accuracy | Speedup |
|---|
(FPCore (x) :precision binary64 (let* ((t_0 (sin (* x 0.5)))) (/ (* (* (/ 8.0 3.0) t_0) t_0) (sin x))))
double code(double x) {
double t_0 = sin((x * 0.5));
return (((8.0 / 3.0) * t_0) * t_0) / sin(x);
}
real(8) function code(x)
real(8), intent (in) :: x
real(8) :: t_0
t_0 = sin((x * 0.5d0))
code = (((8.0d0 / 3.0d0) * t_0) * t_0) / sin(x)
end function
public static double code(double x) {
double t_0 = Math.sin((x * 0.5));
return (((8.0 / 3.0) * t_0) * t_0) / Math.sin(x);
}
def code(x): t_0 = math.sin((x * 0.5)) return (((8.0 / 3.0) * t_0) * t_0) / math.sin(x)
function code(x) t_0 = sin(Float64(x * 0.5)) return Float64(Float64(Float64(Float64(8.0 / 3.0) * t_0) * t_0) / sin(x)) end
function tmp = code(x) t_0 = sin((x * 0.5)); tmp = (((8.0 / 3.0) * t_0) * t_0) / sin(x); end
code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[(8.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
t_0 := \sin \left(x \cdot 0.5\right)\\
\frac{\left(\frac{8}{3} \cdot t_0\right) \cdot t_0}{\sin x}
\end{array}
\end{array}
(FPCore (x) :precision binary64 (* 1.3333333333333333 (tan (/ x 2.0))))
double code(double x) {
return 1.3333333333333333 * tan((x / 2.0));
}
real(8) function code(x)
real(8), intent (in) :: x
code = 1.3333333333333333d0 * tan((x / 2.0d0))
end function
public static double code(double x) {
return 1.3333333333333333 * Math.tan((x / 2.0));
}
def code(x): return 1.3333333333333333 * math.tan((x / 2.0))
function code(x) return Float64(1.3333333333333333 * tan(Float64(x / 2.0))) end
function tmp = code(x) tmp = 1.3333333333333333 * tan((x / 2.0)); end
code[x_] := N[(1.3333333333333333 * N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
\\
1.3333333333333333 \cdot \tan \left(\frac{x}{2}\right)
\end{array}
Initial program 79.0%
associate-*r/99.2%
associate-*l*99.2%
metadata-eval99.2%
Simplified99.2%
associate-*r/79.0%
associate-*r/78.9%
expm1-log1p-u60.3%
associate-*l/60.3%
expm1-udef37.4%
Applied egg-rr36.7%
expm1-def36.7%
expm1-log1p55.4%
associate-*r/55.4%
associate-*l/55.5%
metadata-eval55.5%
distribute-lft-neg-in55.5%
metadata-eval55.5%
distribute-rgt-neg-in55.5%
distribute-lft-in55.5%
sub-neg55.5%
cos-055.5%
metadata-eval55.5%
associate-/r/55.4%
associate-/l*55.5%
*-lft-identity55.5%
times-frac55.4%
*-commutative55.4%
times-frac55.4%
metadata-eval55.4%
Simplified99.5%
Final simplification99.5%
(FPCore (x) :precision binary64 (/ 2.6666666666666665 (+ (* x -0.3333333333333333) (* 4.0 (/ 1.0 x)))))
double code(double x) {
return 2.6666666666666665 / ((x * -0.3333333333333333) + (4.0 * (1.0 / x)));
}
real(8) function code(x)
real(8), intent (in) :: x
code = 2.6666666666666665d0 / ((x * (-0.3333333333333333d0)) + (4.0d0 * (1.0d0 / x)))
end function
public static double code(double x) {
return 2.6666666666666665 / ((x * -0.3333333333333333) + (4.0 * (1.0 / x)));
}
def code(x): return 2.6666666666666665 / ((x * -0.3333333333333333) + (4.0 * (1.0 / x)))
function code(x) return Float64(2.6666666666666665 / Float64(Float64(x * -0.3333333333333333) + Float64(4.0 * Float64(1.0 / x)))) end
function tmp = code(x) tmp = 2.6666666666666665 / ((x * -0.3333333333333333) + (4.0 * (1.0 / x))); end
code[x_] := N[(2.6666666666666665 / N[(N[(x * -0.3333333333333333), $MachinePrecision] + N[(4.0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
\\
\frac{2.6666666666666665}{x \cdot -0.3333333333333333 + 4 \cdot \frac{1}{x}}
\end{array}
Initial program 79.0%
associate-*l*78.9%
associate-*l/78.9%
metadata-eval78.9%
Simplified78.9%
sin-mult55.5%
frac-2neg55.5%
+-inverses55.5%
cos-sum55.2%
cos-255.5%
*-commutative55.5%
associate-*r*55.5%
metadata-eval55.5%
*-un-lft-identity55.5%
metadata-eval55.5%
Applied egg-rr55.5%
neg-sub055.5%
cos-055.5%
associate--r-55.5%
metadata-eval55.5%
Simplified55.5%
*-commutative55.5%
clear-num55.4%
frac-times55.4%
metadata-eval55.4%
Applied egg-rr55.4%
Taylor expanded in x around 0 48.7%
Final simplification48.7%
(FPCore (x) :precision binary64 (* x 0.6666666666666666))
double code(double x) {
return x * 0.6666666666666666;
}
real(8) function code(x)
real(8), intent (in) :: x
code = x * 0.6666666666666666d0
end function
public static double code(double x) {
return x * 0.6666666666666666;
}
def code(x): return x * 0.6666666666666666
function code(x) return Float64(x * 0.6666666666666666) end
function tmp = code(x) tmp = x * 0.6666666666666666; end
code[x_] := N[(x * 0.6666666666666666), $MachinePrecision]
\begin{array}{l}
\\
x \cdot 0.6666666666666666
\end{array}
Initial program 79.0%
associate-*r/99.2%
associate-*l*99.2%
metadata-eval99.2%
Simplified99.2%
Taylor expanded in x around 0 48.2%
Final simplification48.2%
(FPCore (x) :precision binary64 (let* ((t_0 (sin (* x 0.5)))) (/ (/ (* 8.0 t_0) 3.0) (/ (sin x) t_0))))
double code(double x) {
double t_0 = sin((x * 0.5));
return ((8.0 * t_0) / 3.0) / (sin(x) / t_0);
}
real(8) function code(x)
real(8), intent (in) :: x
real(8) :: t_0
t_0 = sin((x * 0.5d0))
code = ((8.0d0 * t_0) / 3.0d0) / (sin(x) / t_0)
end function
public static double code(double x) {
double t_0 = Math.sin((x * 0.5));
return ((8.0 * t_0) / 3.0) / (Math.sin(x) / t_0);
}
def code(x): t_0 = math.sin((x * 0.5)) return ((8.0 * t_0) / 3.0) / (math.sin(x) / t_0)
function code(x) t_0 = sin(Float64(x * 0.5)) return Float64(Float64(Float64(8.0 * t_0) / 3.0) / Float64(sin(x) / t_0)) end
function tmp = code(x) t_0 = sin((x * 0.5)); tmp = ((8.0 * t_0) / 3.0) / (sin(x) / t_0); end
code[x_] := Block[{t$95$0 = N[Sin[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(8.0 * t$95$0), $MachinePrecision] / 3.0), $MachinePrecision] / N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]
\begin{array}{l}
\\
\begin{array}{l}
t_0 := \sin \left(x \cdot 0.5\right)\\
\frac{\frac{8 \cdot t_0}{3}}{\frac{\sin x}{t_0}}
\end{array}
\end{array}
herbie shell --seed 2023224
(FPCore (x)
:name "Graphics.Rasterific.Svg.PathConverter:segmentToBezier from rasterific-svg-0.2.3.1, A"
:precision binary64
:herbie-target
(/ (/ (* 8.0 (sin (* x 0.5))) 3.0) (/ (sin x) (sin (* x 0.5))))
(/ (* (* (/ 8.0 3.0) (sin (* x 0.5))) (sin (* x 0.5))) (sin x)))