
(FPCore (x y z) :precision binary64 (/ (* x (- y z)) y))
double code(double x, double y, double z) {
return (x * (y - z)) / y;
}
real(8) function code(x, y, z)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8), intent (in) :: z
code = (x * (y - z)) / y
end function
public static double code(double x, double y, double z) {
return (x * (y - z)) / y;
}
def code(x, y, z): return (x * (y - z)) / y
function code(x, y, z) return Float64(Float64(x * Float64(y - z)) / y) end
function tmp = code(x, y, z) tmp = (x * (y - z)) / y; end
code[x_, y_, z_] := N[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]
\begin{array}{l}
\\
\frac{x \cdot \left(y - z\right)}{y}
\end{array}
Sampling outcomes in binary64 precision:
Herbie found 6 alternatives:
| Alternative | Accuracy | Speedup |
|---|
(FPCore (x y z) :precision binary64 (/ (* x (- y z)) y))
double code(double x, double y, double z) {
return (x * (y - z)) / y;
}
real(8) function code(x, y, z)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8), intent (in) :: z
code = (x * (y - z)) / y
end function
public static double code(double x, double y, double z) {
return (x * (y - z)) / y;
}
def code(x, y, z): return (x * (y - z)) / y
function code(x, y, z) return Float64(Float64(x * Float64(y - z)) / y) end
function tmp = code(x, y, z) tmp = (x * (y - z)) / y; end
code[x_, y_, z_] := N[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]
\begin{array}{l}
\\
\frac{x \cdot \left(y - z\right)}{y}
\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
(*
x_s
(if (<= x_m 1.35e-230)
(fma (* z x_m) (/ -1.0 y) x_m)
(* (/ (- y z) y) x_m))))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 <= 1.35e-230) {
tmp = fma((z * x_m), (-1.0 / y), x_m);
} else {
tmp = ((y - z) / y) * x_m;
}
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 (x_m <= 1.35e-230) tmp = fma(Float64(z * x_m), Float64(-1.0 / y), x_m); else tmp = Float64(Float64(Float64(y - z) / y) * x_m); end return Float64(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[x$95$m, 1.35e-230], N[(N[(z * x$95$m), $MachinePrecision] * N[(-1.0 / y), $MachinePrecision] + x$95$m), $MachinePrecision], N[(N[(N[(y - z), $MachinePrecision] / y), $MachinePrecision] * x$95$m), $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}\;x\_m \leq 1.35 \cdot 10^{-230}:\\
\;\;\;\;\mathsf{fma}\left(z \cdot x\_m, \frac{-1}{y}, x\_m\right)\\
\mathbf{else}:\\
\;\;\;\;\frac{y - z}{y} \cdot x\_m\\
\end{array}
\end{array}
if x < 1.35000000000000006e-230Initial program 90.3%
lift-/.f64N/A
lift-*.f64N/A
associate-/l*N/A
clear-numN/A
un-div-invN/A
lower-/.f64N/A
lower-/.f6484.1
Applied rewrites84.1%
lift-/.f64N/A
lift-/.f64N/A
associate-/r/N/A
lift--.f64N/A
sub-negN/A
lift-neg.f64N/A
+-commutativeN/A
distribute-lft-inN/A
*-commutativeN/A
clear-numN/A
associate-/r/N/A
associate-*r*N/A
div-invN/A
*-inversesN/A
*-lft-identityN/A
lower-fma.f64N/A
lower-/.f6491.0
Applied rewrites91.0%
lift-fma.f64N/A
lift-/.f64N/A
associate-*l/N/A
frac-2negN/A
div-invN/A
lower-fma.f64N/A
*-commutativeN/A
distribute-lft-neg-inN/A
lift-neg.f64N/A
remove-double-negN/A
lower-*.f64N/A
neg-mul-1N/A
associate-/r*N/A
metadata-evalN/A
lower-/.f6497.3
Applied rewrites97.3%
if 1.35000000000000006e-230 < x Initial program 84.2%
lift-/.f64N/A
lift-*.f64N/A
associate-/l*N/A
*-commutativeN/A
lower-*.f64N/A
lower-/.f6499.9
Applied rewrites99.9%
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 (/ (* (- y z) x_m) y)) (t_1 (* (/ x_m y) (- y z)))) (* x_s (if (<= t_0 0.0) t_1 (if (<= t_0 5e-128) (* 1.0 x_m) t_1)))))
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 = ((y - z) * x_m) / y;
double t_1 = (x_m / y) * (y - z);
double tmp;
if (t_0 <= 0.0) {
tmp = t_1;
} else if (t_0 <= 5e-128) {
tmp = 1.0 * x_m;
} else {
tmp = t_1;
}
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) :: t_1
real(8) :: tmp
t_0 = ((y - z) * x_m) / y
t_1 = (x_m / y) * (y - z)
if (t_0 <= 0.0d0) then
tmp = t_1
else if (t_0 <= 5d-128) then
tmp = 1.0d0 * x_m
else
tmp = t_1
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 = ((y - z) * x_m) / y;
double t_1 = (x_m / y) * (y - z);
double tmp;
if (t_0 <= 0.0) {
tmp = t_1;
} else if (t_0 <= 5e-128) {
tmp = 1.0 * x_m;
} else {
tmp = t_1;
}
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 = ((y - z) * x_m) / y t_1 = (x_m / y) * (y - z) tmp = 0 if t_0 <= 0.0: tmp = t_1 elif t_0 <= 5e-128: tmp = 1.0 * x_m else: tmp = t_1 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(Float64(Float64(y - z) * x_m) / y) t_1 = Float64(Float64(x_m / y) * Float64(y - z)) tmp = 0.0 if (t_0 <= 0.0) tmp = t_1; elseif (t_0 <= 5e-128) tmp = Float64(1.0 * x_m); else tmp = t_1; 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 = ((y - z) * x_m) / y; t_1 = (x_m / y) * (y - z); tmp = 0.0; if (t_0 <= 0.0) tmp = t_1; elseif (t_0 <= 5e-128) tmp = 1.0 * x_m; else tmp = t_1; 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[(N[(N[(y - z), $MachinePrecision] * x$95$m), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$95$m / y), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 5e-128], N[(1.0 * x$95$m), $MachinePrecision], t$95$1]]), $MachinePrecision]]]
\begin{array}{l}
x\_m = \left|x\right|
\\
x\_s = \mathsf{copysign}\left(1, x\right)
\\
\begin{array}{l}
t_0 := \frac{\left(y - z\right) \cdot x\_m}{y}\\
t_1 := \frac{x\_m}{y} \cdot \left(y - z\right)\\
x\_s \cdot \begin{array}{l}
\mathbf{if}\;t\_0 \leq 0:\\
\;\;\;\;t\_1\\
\mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-128}:\\
\;\;\;\;1 \cdot x\_m\\
\mathbf{else}:\\
\;\;\;\;t\_1\\
\end{array}
\end{array}
\end{array}
if (/.f64 (*.f64 x (-.f64 y z)) y) < 0.0 or 5.0000000000000001e-128 < (/.f64 (*.f64 x (-.f64 y z)) y) Initial program 82.8%
lift-/.f64N/A
lift-*.f64N/A
*-commutativeN/A
associate-/l*N/A
*-commutativeN/A
lower-*.f64N/A
lower-/.f6491.8
Applied rewrites91.8%
if 0.0 < (/.f64 (*.f64 x (-.f64 y z)) y) < 5.0000000000000001e-128Initial program 98.7%
lift-/.f64N/A
lift-*.f64N/A
associate-/l*N/A
*-commutativeN/A
lower-*.f64N/A
lower-/.f6499.9
Applied rewrites99.9%
Taylor expanded in y around inf
Applied rewrites90.7%
Final simplification91.7%
(FPCore (x y z) :precision binary64 (if (< z -2.060202331921739e+104) (- x (/ (* z x) y)) (if (< z 1.6939766013828526e+213) (/ x (/ y (- y z))) (* (- y z) (/ x y)))))
double code(double x, double y, double z) {
double tmp;
if (z < -2.060202331921739e+104) {
tmp = x - ((z * x) / y);
} else if (z < 1.6939766013828526e+213) {
tmp = x / (y / (y - z));
} else {
tmp = (y - z) * (x / y);
}
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) :: tmp
if (z < (-2.060202331921739d+104)) then
tmp = x - ((z * x) / y)
else if (z < 1.6939766013828526d+213) then
tmp = x / (y / (y - z))
else
tmp = (y - z) * (x / y)
end if
code = tmp
end function
public static double code(double x, double y, double z) {
double tmp;
if (z < -2.060202331921739e+104) {
tmp = x - ((z * x) / y);
} else if (z < 1.6939766013828526e+213) {
tmp = x / (y / (y - z));
} else {
tmp = (y - z) * (x / y);
}
return tmp;
}
def code(x, y, z): tmp = 0 if z < -2.060202331921739e+104: tmp = x - ((z * x) / y) elif z < 1.6939766013828526e+213: tmp = x / (y / (y - z)) else: tmp = (y - z) * (x / y) return tmp
function code(x, y, z) tmp = 0.0 if (z < -2.060202331921739e+104) tmp = Float64(x - Float64(Float64(z * x) / y)); elseif (z < 1.6939766013828526e+213) tmp = Float64(x / Float64(y / Float64(y - z))); else tmp = Float64(Float64(y - z) * Float64(x / y)); end return tmp end
function tmp_2 = code(x, y, z) tmp = 0.0; if (z < -2.060202331921739e+104) tmp = x - ((z * x) / y); elseif (z < 1.6939766013828526e+213) tmp = x / (y / (y - z)); else tmp = (y - z) * (x / y); end tmp_2 = tmp; end
code[x_, y_, z_] := If[Less[z, -2.060202331921739e+104], N[(x - N[(N[(z * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[Less[z, 1.6939766013828526e+213], N[(x / N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;z < -2.060202331921739 \cdot 10^{+104}:\\
\;\;\;\;x - \frac{z \cdot x}{y}\\
\mathbf{elif}\;z < 1.6939766013828526 \cdot 10^{+213}:\\
\;\;\;\;\frac{x}{\frac{y}{y - z}}\\
\mathbf{else}:\\
\;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\
\end{array}
\end{array}
herbie shell --seed 2024230
(FPCore (x y z)
:name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"
:precision binary64
:alt
(! :herbie-platform default (if (< z -206020233192173900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- x (/ (* z x) y)) (if (< z 1693976601382852600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ x (/ y (- y z))) (* (- y z) (/ x y)))))
(/ (* x (- y z)) y))