
(FPCore (x y z t) :precision binary64 (* x (/ (* (/ y z) t) t)))
double code(double x, double y, double z, double t) {
return x * (((y / z) * t) / t);
}
real(8) function code(x, y, z, t)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8), intent (in) :: z
real(8), intent (in) :: t
code = x * (((y / z) * t) / t)
end function
public static double code(double x, double y, double z, double t) {
return x * (((y / z) * t) / t);
}
def code(x, y, z, t): return x * (((y / z) * t) / t)
function code(x, y, z, t) return Float64(x * Float64(Float64(Float64(y / z) * t) / t)) end
function tmp = code(x, y, z, t) tmp = x * (((y / z) * t) / t); end
code[x_, y_, z_, t_] := N[(x * N[(N[(N[(y / z), $MachinePrecision] * t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
\\
x \cdot \frac{\frac{y}{z} \cdot t}{t}
\end{array}
Sampling outcomes in binary64 precision:
Herbie found 2 alternatives:
| Alternative | Accuracy | Speedup |
|---|
(FPCore (x y z t) :precision binary64 (* x (/ (* (/ y z) t) t)))
double code(double x, double y, double z, double t) {
return x * (((y / z) * t) / t);
}
real(8) function code(x, y, z, t)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8), intent (in) :: z
real(8), intent (in) :: t
code = x * (((y / z) * t) / t)
end function
public static double code(double x, double y, double z, double t) {
return x * (((y / z) * t) / t);
}
def code(x, y, z, t): return x * (((y / z) * t) / t)
function code(x, y, z, t) return Float64(x * Float64(Float64(Float64(y / z) * t) / t)) end
function tmp = code(x, y, z, t) tmp = x * (((y / z) * t) / t); end
code[x_, y_, z_, t_] := N[(x * N[(N[(N[(y / z), $MachinePrecision] * t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
\\
x \cdot \frac{\frac{y}{z} \cdot t}{t}
\end{array}
x\_m = (fabs.f64 x) x\_s = (copysign.f64 #s(literal 1 binary64) x) y\_m = (fabs.f64 y) y\_s = (copysign.f64 #s(literal 1 binary64) y) z\_m = (fabs.f64 z) z\_s = (copysign.f64 #s(literal 1 binary64) z) NOTE: x_m, y_m, z_m, and t should be sorted in increasing order before calling this function. (FPCore (z_s y_s x_s x_m y_m z_m t) :precision binary64 (* z_s (* y_s (* x_s (if (<= z_m 6e-51) (* (/ x_m z_m) y_m) (* x_m (/ y_m z_m)))))))
x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t) {
double tmp;
if (z_m <= 6e-51) {
tmp = (x_m / z_m) * y_m;
} else {
tmp = x_m * (y_m / z_m);
}
return z_s * (y_s * (x_s * tmp));
}
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, and t should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t)
real(8), intent (in) :: z_s
real(8), intent (in) :: y_s
real(8), intent (in) :: x_s
real(8), intent (in) :: x_m
real(8), intent (in) :: y_m
real(8), intent (in) :: z_m
real(8), intent (in) :: t
real(8) :: tmp
if (z_m <= 6d-51) then
tmp = (x_m / z_m) * y_m
else
tmp = x_m * (y_m / z_m)
end if
code = z_s * (y_s * (x_s * tmp))
end function
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t) {
double tmp;
if (z_m <= 6e-51) {
tmp = (x_m / z_m) * y_m;
} else {
tmp = x_m * (y_m / z_m);
}
return z_s * (y_s * (x_s * tmp));
}
x\_m = math.fabs(x) x\_s = math.copysign(1.0, x) y\_m = math.fabs(y) y\_s = math.copysign(1.0, y) z\_m = math.fabs(z) z\_s = math.copysign(1.0, z) [x_m, y_m, z_m, t] = sort([x_m, y_m, z_m, t]) def code(z_s, y_s, x_s, x_m, y_m, z_m, t): tmp = 0 if z_m <= 6e-51: tmp = (x_m / z_m) * y_m else: tmp = x_m * (y_m / z_m) return z_s * (y_s * (x_s * tmp))
x\_m = abs(x) x\_s = copysign(1.0, x) y\_m = abs(y) y\_s = copysign(1.0, y) z\_m = abs(z) z\_s = copysign(1.0, z) x_m, y_m, z_m, t = sort([x_m, y_m, z_m, t]) function code(z_s, y_s, x_s, x_m, y_m, z_m, t) tmp = 0.0 if (z_m <= 6e-51) tmp = Float64(Float64(x_m / z_m) * y_m); else tmp = Float64(x_m * Float64(y_m / z_m)); end return Float64(z_s * Float64(y_s * Float64(x_s * tmp))) end
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t = num2cell(sort([x_m, y_m, z_m, t])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t)
tmp = 0.0;
if (z_m <= 6e-51)
tmp = (x_m / z_m) * y_m;
else
tmp = x_m * (y_m / z_m);
end
tmp_2 = z_s * (y_s * (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]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, and t should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 6e-51], N[(N[(x$95$m / z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], N[(x$95$m * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
x\_m = \left|x\right|
\\
x\_s = \mathsf{copysign}\left(1, x\right)
\\
y\_m = \left|y\right|
\\
y\_s = \mathsf{copysign}\left(1, y\right)
\\
z\_m = \left|z\right|
\\
z\_s = \mathsf{copysign}\left(1, z\right)
\\
[x_m, y_m, z_m, t] = \mathsf{sort}([x_m, y_m, z_m, t])\\
\\
z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l}
\mathbf{if}\;z\_m \leq 6 \cdot 10^{-51}:\\
\;\;\;\;\frac{x\_m}{z\_m} \cdot y\_m\\
\mathbf{else}:\\
\;\;\;\;x\_m \cdot \frac{y\_m}{z\_m}\\
\end{array}\right)\right)
\end{array}
if z < 6.00000000000000005e-51Initial program 79.7%
clear-numN/A
un-div-invN/A
*-commutativeN/A
associate-/r*N/A
*-inversesN/A
clear-numN/A
associate-/r/N/A
*-lowering-*.f64N/A
/-lowering-/.f6492.0%
Applied egg-rr92.0%
if 6.00000000000000005e-51 < z Initial program 90.4%
associate-*r/N/A
associate-*r*N/A
associate-/l*N/A
*-inversesN/A
*-rgt-identityN/A
*-commutativeN/A
*-lowering-*.f64N/A
/-lowering-/.f6498.0%
Applied egg-rr98.0%
Final simplification93.8%
x\_m = (fabs.f64 x) x\_s = (copysign.f64 #s(literal 1 binary64) x) y\_m = (fabs.f64 y) y\_s = (copysign.f64 #s(literal 1 binary64) y) z\_m = (fabs.f64 z) z\_s = (copysign.f64 #s(literal 1 binary64) z) NOTE: x_m, y_m, z_m, and t should be sorted in increasing order before calling this function. (FPCore (z_s y_s x_s x_m y_m z_m t) :precision binary64 (* z_s (* y_s (* x_s (* (/ x_m z_m) y_m)))))
x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t) {
return z_s * (y_s * (x_s * ((x_m / z_m) * y_m)));
}
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, and t should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t)
real(8), intent (in) :: z_s
real(8), intent (in) :: y_s
real(8), intent (in) :: x_s
real(8), intent (in) :: x_m
real(8), intent (in) :: y_m
real(8), intent (in) :: z_m
real(8), intent (in) :: t
code = z_s * (y_s * (x_s * ((x_m / z_m) * y_m)))
end function
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t) {
return z_s * (y_s * (x_s * ((x_m / z_m) * y_m)));
}
x\_m = math.fabs(x) x\_s = math.copysign(1.0, x) y\_m = math.fabs(y) y\_s = math.copysign(1.0, y) z\_m = math.fabs(z) z\_s = math.copysign(1.0, z) [x_m, y_m, z_m, t] = sort([x_m, y_m, z_m, t]) def code(z_s, y_s, x_s, x_m, y_m, z_m, t): return z_s * (y_s * (x_s * ((x_m / z_m) * y_m)))
x\_m = abs(x) x\_s = copysign(1.0, x) y\_m = abs(y) y\_s = copysign(1.0, y) z\_m = abs(z) z\_s = copysign(1.0, z) x_m, y_m, z_m, t = sort([x_m, y_m, z_m, t]) function code(z_s, y_s, x_s, x_m, y_m, z_m, t) return Float64(z_s * Float64(y_s * Float64(x_s * Float64(Float64(x_m / z_m) * y_m)))) end
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t = num2cell(sort([x_m, y_m, z_m, t])){:}
function tmp = code(z_s, y_s, x_s, x_m, y_m, z_m, t)
tmp = z_s * (y_s * (x_s * ((x_m / z_m) * y_m)));
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]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, and t should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * N[(N[(x$95$m / z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
x\_m = \left|x\right|
\\
x\_s = \mathsf{copysign}\left(1, x\right)
\\
y\_m = \left|y\right|
\\
y\_s = \mathsf{copysign}\left(1, y\right)
\\
z\_m = \left|z\right|
\\
z\_s = \mathsf{copysign}\left(1, z\right)
\\
[x_m, y_m, z_m, t] = \mathsf{sort}([x_m, y_m, z_m, t])\\
\\
z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \left(\frac{x\_m}{z\_m} \cdot y\_m\right)\right)\right)
\end{array}
Initial program 82.8%
clear-numN/A
un-div-invN/A
*-commutativeN/A
associate-/r*N/A
*-inversesN/A
clear-numN/A
associate-/r/N/A
*-lowering-*.f64N/A
/-lowering-/.f6491.8%
Applied egg-rr91.8%
(FPCore (x y z t)
:precision binary64
(let* ((t_1 (* x (/ y z))) (t_2 (/ (* (/ y z) t) t)) (t_3 (/ y (/ z x))))
(if (< t_2 -1.20672205123045e+245)
t_3
(if (< t_2 -5.907522236933906e-275)
t_1
(if (< t_2 5.658954423153415e-65)
t_3
(if (< t_2 2.0087180502407133e+217) t_1 (/ (* y x) z)))))))
double code(double x, double y, double z, double t) {
double t_1 = x * (y / z);
double t_2 = ((y / z) * t) / t;
double t_3 = y / (z / x);
double tmp;
if (t_2 < -1.20672205123045e+245) {
tmp = t_3;
} else if (t_2 < -5.907522236933906e-275) {
tmp = t_1;
} else if (t_2 < 5.658954423153415e-65) {
tmp = t_3;
} else if (t_2 < 2.0087180502407133e+217) {
tmp = t_1;
} else {
tmp = (y * x) / z;
}
return tmp;
}
real(8) function code(x, y, z, t)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8), intent (in) :: z
real(8), intent (in) :: t
real(8) :: t_1
real(8) :: t_2
real(8) :: t_3
real(8) :: tmp
t_1 = x * (y / z)
t_2 = ((y / z) * t) / t
t_3 = y / (z / x)
if (t_2 < (-1.20672205123045d+245)) then
tmp = t_3
else if (t_2 < (-5.907522236933906d-275)) then
tmp = t_1
else if (t_2 < 5.658954423153415d-65) then
tmp = t_3
else if (t_2 < 2.0087180502407133d+217) then
tmp = t_1
else
tmp = (y * x) / z
end if
code = tmp
end function
public static double code(double x, double y, double z, double t) {
double t_1 = x * (y / z);
double t_2 = ((y / z) * t) / t;
double t_3 = y / (z / x);
double tmp;
if (t_2 < -1.20672205123045e+245) {
tmp = t_3;
} else if (t_2 < -5.907522236933906e-275) {
tmp = t_1;
} else if (t_2 < 5.658954423153415e-65) {
tmp = t_3;
} else if (t_2 < 2.0087180502407133e+217) {
tmp = t_1;
} else {
tmp = (y * x) / z;
}
return tmp;
}
def code(x, y, z, t): t_1 = x * (y / z) t_2 = ((y / z) * t) / t t_3 = y / (z / x) tmp = 0 if t_2 < -1.20672205123045e+245: tmp = t_3 elif t_2 < -5.907522236933906e-275: tmp = t_1 elif t_2 < 5.658954423153415e-65: tmp = t_3 elif t_2 < 2.0087180502407133e+217: tmp = t_1 else: tmp = (y * x) / z return tmp
function code(x, y, z, t) t_1 = Float64(x * Float64(y / z)) t_2 = Float64(Float64(Float64(y / z) * t) / t) t_3 = Float64(y / Float64(z / x)) tmp = 0.0 if (t_2 < -1.20672205123045e+245) tmp = t_3; elseif (t_2 < -5.907522236933906e-275) tmp = t_1; elseif (t_2 < 5.658954423153415e-65) tmp = t_3; elseif (t_2 < 2.0087180502407133e+217) tmp = t_1; else tmp = Float64(Float64(y * x) / z); end return tmp end
function tmp_2 = code(x, y, z, t) t_1 = x * (y / z); t_2 = ((y / z) * t) / t; t_3 = y / (z / x); tmp = 0.0; if (t_2 < -1.20672205123045e+245) tmp = t_3; elseif (t_2 < -5.907522236933906e-275) tmp = t_1; elseif (t_2 < 5.658954423153415e-65) tmp = t_3; elseif (t_2 < 2.0087180502407133e+217) tmp = t_1; else tmp = (y * x) / z; end tmp_2 = tmp; end
code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y / z), $MachinePrecision] * t), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$3 = N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$2, -1.20672205123045e+245], t$95$3, If[Less[t$95$2, -5.907522236933906e-275], t$95$1, If[Less[t$95$2, 5.658954423153415e-65], t$95$3, If[Less[t$95$2, 2.0087180502407133e+217], t$95$1, N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]]]]]]]]
\begin{array}{l}
\\
\begin{array}{l}
t_1 := x \cdot \frac{y}{z}\\
t_2 := \frac{\frac{y}{z} \cdot t}{t}\\
t_3 := \frac{y}{\frac{z}{x}}\\
\mathbf{if}\;t\_2 < -1.20672205123045 \cdot 10^{+245}:\\
\;\;\;\;t\_3\\
\mathbf{elif}\;t\_2 < -5.907522236933906 \cdot 10^{-275}:\\
\;\;\;\;t\_1\\
\mathbf{elif}\;t\_2 < 5.658954423153415 \cdot 10^{-65}:\\
\;\;\;\;t\_3\\
\mathbf{elif}\;t\_2 < 2.0087180502407133 \cdot 10^{+217}:\\
\;\;\;\;t\_1\\
\mathbf{else}:\\
\;\;\;\;\frac{y \cdot x}{z}\\
\end{array}
\end{array}
herbie shell --seed 2024191
(FPCore (x y z t)
:name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B"
:precision binary64
:alt
(! :herbie-platform default (if (< (/ (* (/ y z) t) t) -120672205123045000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) -2953761118466953/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* x (/ y z)) (if (< (/ (* (/ y z) t) t) 1131790884630683/20000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) 20087180502407133000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* x (/ y z)) (/ (* y x) z))))))
(* x (/ (* (/ y z) t) t)))