
(FPCore (x y z) :precision binary64 (/ (* x (/ (sin y) y)) z))
double code(double x, double y, double z) {
return (x * (sin(y) / y)) / z;
}
real(8) function code(x, y, z)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8), intent (in) :: z
code = (x * (sin(y) / y)) / z
end function
public static double code(double x, double y, double z) {
return (x * (Math.sin(y) / y)) / z;
}
def code(x, y, z): return (x * (math.sin(y) / y)) / z
function code(x, y, z) return Float64(Float64(x * Float64(sin(y) / y)) / z) end
function tmp = code(x, y, z) tmp = (x * (sin(y) / y)) / z; end
code[x_, y_, z_] := N[(N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]
\begin{array}{l}
\\
\frac{x \cdot \frac{\sin y}{y}}{z}
\end{array}
Sampling outcomes in binary64 precision:
Herbie found 4 alternatives:
| Alternative | Accuracy | Speedup |
|---|
(FPCore (x y z) :precision binary64 (/ (* x (/ (sin y) y)) z))
double code(double x, double y, double z) {
return (x * (sin(y) / y)) / z;
}
real(8) function code(x, y, z)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8), intent (in) :: z
code = (x * (sin(y) / y)) / z
end function
public static double code(double x, double y, double z) {
return (x * (Math.sin(y) / y)) / z;
}
def code(x, y, z): return (x * (math.sin(y) / y)) / z
function code(x, y, z) return Float64(Float64(x * Float64(sin(y) / y)) / z) end
function tmp = code(x, y, z) tmp = (x * (sin(y) / y)) / z; end
code[x_, y_, z_] := N[(N[(x * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]
\begin{array}{l}
\\
\frac{x \cdot \frac{\sin y}{y}}{z}
\end{array}
x\_m = (fabs.f64 x) x\_s = (copysign.f64 #s(literal 1 binary64) x) (FPCore (x_s x_m y z) :precision binary64 (let* ((t_0 (* x_m (/ (sin y) y)))) (* x_s (if (<= t_0 0.0) (/ (* x_m (sin y)) (* y z)) (/ t_0 z)))))
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
double t_0 = x_m * (sin(y) / y);
double tmp;
if (t_0 <= 0.0) {
tmp = (x_m * sin(y)) / (y * z);
} else {
tmp = t_0 / z;
}
return x_s * tmp;
}
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
real(8), intent (in) :: x_s
real(8), intent (in) :: x_m
real(8), intent (in) :: y
real(8), intent (in) :: z
real(8) :: t_0
real(8) :: tmp
t_0 = x_m * (sin(y) / y)
if (t_0 <= 0.0d0) then
tmp = (x_m * sin(y)) / (y * z)
else
tmp = t_0 / z
end if
code = x_s * tmp
end function
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
double t_0 = x_m * (Math.sin(y) / y);
double tmp;
if (t_0 <= 0.0) {
tmp = (x_m * Math.sin(y)) / (y * z);
} else {
tmp = t_0 / z;
}
return x_s * tmp;
}
x\_m = math.fabs(x) x\_s = math.copysign(1.0, x) def code(x_s, x_m, y, z): t_0 = x_m * (math.sin(y) / y) tmp = 0 if t_0 <= 0.0: tmp = (x_m * math.sin(y)) / (y * z) else: tmp = t_0 / z return x_s * tmp
x\_m = abs(x) x\_s = copysign(1.0, x) function code(x_s, x_m, y, z) t_0 = Float64(x_m * Float64(sin(y) / y)) tmp = 0.0 if (t_0 <= 0.0) tmp = Float64(Float64(x_m * sin(y)) / Float64(y * z)); else tmp = Float64(t_0 / z); end return Float64(x_s * tmp) end
x\_m = abs(x); x\_s = sign(x) * abs(1.0); function tmp_2 = code(x_s, x_m, y, z) t_0 = x_m * (sin(y) / y); tmp = 0.0; if (t_0 <= 0.0) tmp = (x_m * sin(y)) / (y * z); else tmp = t_0 / z; end tmp_2 = x_s * tmp; end
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, 0.0], N[(N[(x$95$m * N[Sin[y], $MachinePrecision]), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / z), $MachinePrecision]]), $MachinePrecision]]
\begin{array}{l}
x\_m = \left|x\right|
\\
x\_s = \mathsf{copysign}\left(1, x\right)
\\
\begin{array}{l}
t_0 := x\_m \cdot \frac{\sin y}{y}\\
x\_s \cdot \begin{array}{l}
\mathbf{if}\;t\_0 \leq 0:\\
\;\;\;\;\frac{x\_m \cdot \sin y}{y \cdot z}\\
\mathbf{else}:\\
\;\;\;\;\frac{t\_0}{z}\\
\end{array}
\end{array}
\end{array}
if (*.f64 x (/.f64 (sin.f64 y) y)) < 0.0Initial program 91.8%
Taylor expanded in x around 0
lower-/.f64N/A
lower-*.f64N/A
lower-sin.f64N/A
lower-*.f6482.8
Simplified82.8%
if 0.0 < (*.f64 x (/.f64 (sin.f64 y) y)) Initial program 99.4%
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
:precision binary64
(*
x_s
(if (<= (/ (sin y) y) 0.9999999999999998)
(/ (* x_m (sin y)) (* y z))
(/ x_m z))))x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
double tmp;
if ((sin(y) / y) <= 0.9999999999999998) {
tmp = (x_m * sin(y)) / (y * z);
} else {
tmp = x_m / z;
}
return x_s * tmp;
}
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
real(8), intent (in) :: x_s
real(8), intent (in) :: x_m
real(8), intent (in) :: y
real(8), intent (in) :: z
real(8) :: tmp
if ((sin(y) / y) <= 0.9999999999999998d0) then
tmp = (x_m * sin(y)) / (y * z)
else
tmp = x_m / z
end if
code = x_s * tmp
end function
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
double tmp;
if ((Math.sin(y) / y) <= 0.9999999999999998) {
tmp = (x_m * Math.sin(y)) / (y * z);
} else {
tmp = x_m / z;
}
return x_s * tmp;
}
x\_m = math.fabs(x) x\_s = math.copysign(1.0, x) def code(x_s, x_m, y, z): tmp = 0 if (math.sin(y) / y) <= 0.9999999999999998: tmp = (x_m * math.sin(y)) / (y * z) else: tmp = x_m / z return x_s * tmp
x\_m = abs(x) x\_s = copysign(1.0, x) function code(x_s, x_m, y, z) tmp = 0.0 if (Float64(sin(y) / y) <= 0.9999999999999998) tmp = Float64(Float64(x_m * sin(y)) / Float64(y * z)); else tmp = Float64(x_m / z); end return Float64(x_s * tmp) end
x\_m = abs(x); x\_s = sign(x) * abs(1.0); function tmp_2 = code(x_s, x_m, y, z) tmp = 0.0; if ((sin(y) / y) <= 0.9999999999999998) tmp = (x_m * sin(y)) / (y * z); else tmp = x_m / z; end tmp_2 = x_s * tmp; end
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], 0.9999999999999998], N[(N[(x$95$m * N[Sin[y], $MachinePrecision]), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]
\begin{array}{l}
x\_m = \left|x\right|
\\
x\_s = \mathsf{copysign}\left(1, x\right)
\\
x\_s \cdot \begin{array}{l}
\mathbf{if}\;\frac{\sin y}{y} \leq 0.9999999999999998:\\
\;\;\;\;\frac{x\_m \cdot \sin y}{y \cdot z}\\
\mathbf{else}:\\
\;\;\;\;\frac{x\_m}{z}\\
\end{array}
\end{array}
if (/.f64 (sin.f64 y) y) < 0.99999999999999978Initial program 90.6%
Taylor expanded in x around 0
lower-/.f64N/A
lower-*.f64N/A
lower-sin.f64N/A
lower-*.f6491.0
Simplified91.0%
if 0.99999999999999978 < (/.f64 (sin.f64 y) y) Initial program 100.0%
Taylor expanded in y around 0
lower-/.f64100.0
Simplified100.0%
x\_m = (fabs.f64 x) x\_s = (copysign.f64 #s(literal 1 binary64) x) (FPCore (x_s x_m y z) :precision binary64 (* x_s (if (<= (/ (* x_m (/ (sin y) y)) z) 0.0) (/ (* x_m y) (* y z)) (/ x_m z))))
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
double tmp;
if (((x_m * (sin(y) / y)) / z) <= 0.0) {
tmp = (x_m * y) / (y * z);
} else {
tmp = x_m / z;
}
return x_s * tmp;
}
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
real(8), intent (in) :: x_s
real(8), intent (in) :: x_m
real(8), intent (in) :: y
real(8), intent (in) :: z
real(8) :: tmp
if (((x_m * (sin(y) / y)) / z) <= 0.0d0) then
tmp = (x_m * y) / (y * z)
else
tmp = x_m / z
end if
code = x_s * tmp
end function
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
double tmp;
if (((x_m * (Math.sin(y) / y)) / z) <= 0.0) {
tmp = (x_m * y) / (y * z);
} else {
tmp = x_m / z;
}
return x_s * tmp;
}
x\_m = math.fabs(x) x\_s = math.copysign(1.0, x) def code(x_s, x_m, y, z): tmp = 0 if ((x_m * (math.sin(y) / y)) / z) <= 0.0: tmp = (x_m * y) / (y * z) else: tmp = x_m / z return x_s * tmp
x\_m = abs(x) x\_s = copysign(1.0, x) function code(x_s, x_m, y, z) tmp = 0.0 if (Float64(Float64(x_m * Float64(sin(y) / y)) / z) <= 0.0) tmp = Float64(Float64(x_m * y) / Float64(y * z)); else tmp = Float64(x_m / z); end return Float64(x_s * tmp) end
x\_m = abs(x); x\_s = sign(x) * abs(1.0); function tmp_2 = code(x_s, x_m, y, z) tmp = 0.0; if (((x_m * (sin(y) / y)) / z) <= 0.0) tmp = (x_m * y) / (y * z); else tmp = x_m / z; end tmp_2 = x_s * tmp; end
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 0.0], N[(N[(x$95$m * y), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / z), $MachinePrecision]]), $MachinePrecision]
\begin{array}{l}
x\_m = \left|x\right|
\\
x\_s = \mathsf{copysign}\left(1, x\right)
\\
x\_s \cdot \begin{array}{l}
\mathbf{if}\;\frac{x\_m \cdot \frac{\sin y}{y}}{z} \leq 0:\\
\;\;\;\;\frac{x\_m \cdot y}{y \cdot z}\\
\mathbf{else}:\\
\;\;\;\;\frac{x\_m}{z}\\
\end{array}
\end{array}
if (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) < 0.0Initial program 92.6%
Taylor expanded in x around 0
lower-/.f64N/A
lower-*.f64N/A
lower-sin.f64N/A
lower-*.f6484.7
Simplified84.7%
Taylor expanded in y around 0
lower-*.f6449.6
Simplified49.6%
if 0.0 < (/.f64 (*.f64 x (/.f64 (sin.f64 y) y)) z) Initial program 99.8%
Taylor expanded in y around 0
lower-/.f6457.8
Simplified57.8%
x\_m = (fabs.f64 x) x\_s = (copysign.f64 #s(literal 1 binary64) x) (FPCore (x_s x_m y z) :precision binary64 (* x_s (/ x_m z)))
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
return x_s * (x_m / z);
}
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
real(8), intent (in) :: x_s
real(8), intent (in) :: x_m
real(8), intent (in) :: y
real(8), intent (in) :: z
code = x_s * (x_m / z)
end function
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
return x_s * (x_m / z);
}
x\_m = math.fabs(x) x\_s = math.copysign(1.0, x) def code(x_s, x_m, y, z): return x_s * (x_m / z)
x\_m = abs(x) x\_s = copysign(1.0, x) function code(x_s, x_m, y, z) return Float64(x_s * Float64(x_m / z)) end
x\_m = abs(x); x\_s = sign(x) * abs(1.0); function tmp = code(x_s, x_m, y, z) tmp = x_s * (x_m / z); end
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
x\_m = \left|x\right|
\\
x\_s = \mathsf{copysign}\left(1, x\right)
\\
x\_s \cdot \frac{x\_m}{z}
\end{array}
Initial program 95.2%
Taylor expanded in y around 0
lower-/.f6458.6
Simplified58.6%
(FPCore (x y z)
:precision binary64
(let* ((t_0 (/ y (sin y))) (t_1 (/ (* x (/ 1.0 t_0)) z)))
(if (< z -4.2173720203427147e-29)
t_1
(if (< z 4.446702369113811e+64) (/ x (* z t_0)) t_1))))
double code(double x, double y, double z) {
double t_0 = y / sin(y);
double t_1 = (x * (1.0 / t_0)) / z;
double tmp;
if (z < -4.2173720203427147e-29) {
tmp = t_1;
} else if (z < 4.446702369113811e+64) {
tmp = x / (z * t_0);
} else {
tmp = t_1;
}
return tmp;
}
real(8) function code(x, y, z)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8), intent (in) :: z
real(8) :: t_0
real(8) :: t_1
real(8) :: tmp
t_0 = y / sin(y)
t_1 = (x * (1.0d0 / t_0)) / z
if (z < (-4.2173720203427147d-29)) then
tmp = t_1
else if (z < 4.446702369113811d+64) then
tmp = x / (z * t_0)
else
tmp = t_1
end if
code = tmp
end function
public static double code(double x, double y, double z) {
double t_0 = y / Math.sin(y);
double t_1 = (x * (1.0 / t_0)) / z;
double tmp;
if (z < -4.2173720203427147e-29) {
tmp = t_1;
} else if (z < 4.446702369113811e+64) {
tmp = x / (z * t_0);
} else {
tmp = t_1;
}
return tmp;
}
def code(x, y, z): t_0 = y / math.sin(y) t_1 = (x * (1.0 / t_0)) / z tmp = 0 if z < -4.2173720203427147e-29: tmp = t_1 elif z < 4.446702369113811e+64: tmp = x / (z * t_0) else: tmp = t_1 return tmp
function code(x, y, z) t_0 = Float64(y / sin(y)) t_1 = Float64(Float64(x * Float64(1.0 / t_0)) / z) tmp = 0.0 if (z < -4.2173720203427147e-29) tmp = t_1; elseif (z < 4.446702369113811e+64) tmp = Float64(x / Float64(z * t_0)); else tmp = t_1; end return tmp end
function tmp_2 = code(x, y, z) t_0 = y / sin(y); t_1 = (x * (1.0 / t_0)) / z; tmp = 0.0; if (z < -4.2173720203427147e-29) tmp = t_1; elseif (z < 4.446702369113811e+64) tmp = x / (z * t_0); else tmp = t_1; end tmp_2 = tmp; end
code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[Less[z, -4.2173720203427147e-29], t$95$1, If[Less[z, 4.446702369113811e+64], N[(x / N[(z * t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]
\begin{array}{l}
\\
\begin{array}{l}
t_0 := \frac{y}{\sin y}\\
t_1 := \frac{x \cdot \frac{1}{t\_0}}{z}\\
\mathbf{if}\;z < -4.2173720203427147 \cdot 10^{-29}:\\
\;\;\;\;t\_1\\
\mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\
\;\;\;\;\frac{x}{z \cdot t\_0}\\
\mathbf{else}:\\
\;\;\;\;t\_1\\
\end{array}
\end{array}
herbie shell --seed 2024215
(FPCore (x y z)
:name "Linear.Quaternion:$ctanh from linear-1.19.1.3"
:precision binary64
:alt
(! :herbie-platform default (if (< z -42173720203427147/1000000000000000000000000000000000000000000000) (/ (* x (/ 1 (/ y (sin y)))) z) (if (< z 44467023691138110000000000000000000000000000000000000000000000000) (/ x (* z (/ y (sin y)))) (/ (* x (/ 1 (/ y (sin y)))) z))))
(/ (* x (/ (sin y) y)) z))