
(FPCore (x y z) :precision binary64 (/ (+ x y) (- 1.0 (/ y z))))
double code(double x, double y, double z) {
return (x + y) / (1.0 - (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 + y) / (1.0d0 - (y / z))
end function
public static double code(double x, double y, double z) {
return (x + y) / (1.0 - (y / z));
}
def code(x, y, z): return (x + y) / (1.0 - (y / z))
function code(x, y, z) return Float64(Float64(x + y) / Float64(1.0 - Float64(y / z))) end
function tmp = code(x, y, z) tmp = (x + y) / (1.0 - (y / z)); end
code[x_, y_, z_] := N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
\\
\frac{x + y}{1 - \frac{y}{z}}
\end{array}
Sampling outcomes in binary64 precision:
Herbie found 4 alternatives:
| Alternative | Accuracy | Speedup |
|---|
(FPCore (x y z) :precision binary64 (/ (+ x y) (- 1.0 (/ y z))))
double code(double x, double y, double z) {
return (x + y) / (1.0 - (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 + y) / (1.0d0 - (y / z))
end function
public static double code(double x, double y, double z) {
return (x + y) / (1.0 - (y / z));
}
def code(x, y, z): return (x + y) / (1.0 - (y / z))
function code(x, y, z) return Float64(Float64(x + y) / Float64(1.0 - Float64(y / z))) end
function tmp = code(x, y, z) tmp = (x + y) / (1.0 - (y / z)); end
code[x_, y_, z_] := N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
\begin{array}{l}
\\
\frac{x + y}{1 - \frac{y}{z}}
\end{array}
(FPCore (x y z) :precision binary64 (let* ((t_0 (/ (+ x y) (- 1.0 (/ y z))))) (if (<= t_0 -5e-304) t_0 (if (<= t_0 0.0) (- (fma z (/ x y) z)) t_0))))
double code(double x, double y, double z) {
double t_0 = (x + y) / (1.0 - (y / z));
double tmp;
if (t_0 <= -5e-304) {
tmp = t_0;
} else if (t_0 <= 0.0) {
tmp = -fma(z, (x / y), z);
} else {
tmp = t_0;
}
return tmp;
}
function code(x, y, z) t_0 = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z))) tmp = 0.0 if (t_0 <= -5e-304) tmp = t_0; elseif (t_0 <= 0.0) tmp = Float64(-fma(z, Float64(x / y), z)); else tmp = t_0; end return tmp end
code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-304], t$95$0, If[LessEqual[t$95$0, 0.0], (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision]), t$95$0]]]
\begin{array}{l}
\\
\begin{array}{l}
t_0 := \frac{x + y}{1 - \frac{y}{z}}\\
\mathbf{if}\;t\_0 \leq -5 \cdot 10^{-304}:\\
\;\;\;\;t\_0\\
\mathbf{elif}\;t\_0 \leq 0:\\
\;\;\;\;-\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\
\mathbf{else}:\\
\;\;\;\;t\_0\\
\end{array}
\end{array}
if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -4.99999999999999965e-304 or -0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) Initial program 99.9%
if -4.99999999999999965e-304 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0Initial program 6.7%
Taylor expanded in z around 0
mul-1-negN/A
associate-/l*N/A
distribute-rgt-neg-inN/A
distribute-neg-fracN/A
+-commutativeN/A
distribute-neg-inN/A
neg-mul-1N/A
unsub-negN/A
div-subN/A
associate-*l/N/A
metadata-evalN/A
distribute-neg-fracN/A
distribute-lft-neg-outN/A
lft-mult-inverseN/A
metadata-evalN/A
distribute-lft-out--N/A
*-commutativeN/A
associate-/l*N/A
*-commutativeN/A
unsub-negN/A
mul-1-negN/A
distribute-neg-outN/A
lower-neg.f64N/A
Simplified99.9%
(FPCore (x y z) :precision binary64 (let* ((t_0 (- (fma z (/ x y) z)))) (if (<= y -2400.0) t_0 (if (<= y 60000000.0) (+ x y) t_0))))
double code(double x, double y, double z) {
double t_0 = -fma(z, (x / y), z);
double tmp;
if (y <= -2400.0) {
tmp = t_0;
} else if (y <= 60000000.0) {
tmp = x + y;
} else {
tmp = t_0;
}
return tmp;
}
function code(x, y, z) t_0 = Float64(-fma(z, Float64(x / y), z)) tmp = 0.0 if (y <= -2400.0) tmp = t_0; elseif (y <= 60000000.0) tmp = Float64(x + y); else tmp = t_0; end return tmp end
code[x_, y_, z_] := Block[{t$95$0 = (-N[(z * N[(x / y), $MachinePrecision] + z), $MachinePrecision])}, If[LessEqual[y, -2400.0], t$95$0, If[LessEqual[y, 60000000.0], N[(x + y), $MachinePrecision], t$95$0]]]
\begin{array}{l}
\\
\begin{array}{l}
t_0 := -\mathsf{fma}\left(z, \frac{x}{y}, z\right)\\
\mathbf{if}\;y \leq -2400:\\
\;\;\;\;t\_0\\
\mathbf{elif}\;y \leq 60000000:\\
\;\;\;\;x + y\\
\mathbf{else}:\\
\;\;\;\;t\_0\\
\end{array}
\end{array}
if y < -2400 or 6e7 < y Initial program 72.9%
Taylor expanded in z around 0
mul-1-negN/A
associate-/l*N/A
distribute-rgt-neg-inN/A
distribute-neg-fracN/A
+-commutativeN/A
distribute-neg-inN/A
neg-mul-1N/A
unsub-negN/A
div-subN/A
associate-*l/N/A
metadata-evalN/A
distribute-neg-fracN/A
distribute-lft-neg-outN/A
lft-mult-inverseN/A
metadata-evalN/A
distribute-lft-out--N/A
*-commutativeN/A
associate-/l*N/A
*-commutativeN/A
unsub-negN/A
mul-1-negN/A
distribute-neg-outN/A
lower-neg.f64N/A
Simplified81.2%
if -2400 < y < 6e7Initial program 99.9%
Taylor expanded in z around inf
+-commutativeN/A
lower-+.f6479.6
Simplified79.6%
Final simplification80.4%
(FPCore (x y z) :precision binary64 (if (<= y -4e+114) (- z) (if (<= y 60000000.0) (+ x y) (- z))))
double code(double x, double y, double z) {
double tmp;
if (y <= -4e+114) {
tmp = -z;
} else if (y <= 60000000.0) {
tmp = x + y;
} else {
tmp = -z;
}
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 (y <= (-4d+114)) then
tmp = -z
else if (y <= 60000000.0d0) then
tmp = x + y
else
tmp = -z
end if
code = tmp
end function
public static double code(double x, double y, double z) {
double tmp;
if (y <= -4e+114) {
tmp = -z;
} else if (y <= 60000000.0) {
tmp = x + y;
} else {
tmp = -z;
}
return tmp;
}
def code(x, y, z): tmp = 0 if y <= -4e+114: tmp = -z elif y <= 60000000.0: tmp = x + y else: tmp = -z return tmp
function code(x, y, z) tmp = 0.0 if (y <= -4e+114) tmp = Float64(-z); elseif (y <= 60000000.0) tmp = Float64(x + y); else tmp = Float64(-z); end return tmp end
function tmp_2 = code(x, y, z) tmp = 0.0; if (y <= -4e+114) tmp = -z; elseif (y <= 60000000.0) tmp = x + y; else tmp = -z; end tmp_2 = tmp; end
code[x_, y_, z_] := If[LessEqual[y, -4e+114], (-z), If[LessEqual[y, 60000000.0], N[(x + y), $MachinePrecision], (-z)]]
\begin{array}{l}
\\
\begin{array}{l}
\mathbf{if}\;y \leq -4 \cdot 10^{+114}:\\
\;\;\;\;-z\\
\mathbf{elif}\;y \leq 60000000:\\
\;\;\;\;x + y\\
\mathbf{else}:\\
\;\;\;\;-z\\
\end{array}
\end{array}
if y < -4e114 or 6e7 < y Initial program 69.4%
Taylor expanded in y around inf
mul-1-negN/A
lower-neg.f6474.9
Simplified74.9%
if -4e114 < y < 6e7Initial program 98.0%
Taylor expanded in z around inf
+-commutativeN/A
lower-+.f6473.8
Simplified73.8%
Final simplification74.3%
(FPCore (x y z) :precision binary64 (- z))
double code(double x, double y, double z) {
return -z;
}
real(8) function code(x, y, z)
real(8), intent (in) :: x
real(8), intent (in) :: y
real(8), intent (in) :: z
code = -z
end function
public static double code(double x, double y, double z) {
return -z;
}
def code(x, y, z): return -z
function code(x, y, z) return Float64(-z) end
function tmp = code(x, y, z) tmp = -z; end
code[x_, y_, z_] := (-z)
\begin{array}{l}
\\
-z
\end{array}
Initial program 86.4%
Taylor expanded in y around inf
mul-1-negN/A
lower-neg.f6438.8
Simplified38.8%
(FPCore (x y z)
:precision binary64
(let* ((t_0 (* (/ (+ y x) (- y)) z)))
(if (< y -3.7429310762689856e+171)
t_0
(if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) t_0))))
double code(double x, double y, double z) {
double t_0 = ((y + x) / -y) * z;
double tmp;
if (y < -3.7429310762689856e+171) {
tmp = t_0;
} else if (y < 3.5534662456086734e+168) {
tmp = (x + y) / (1.0 - (y / z));
} else {
tmp = t_0;
}
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) :: tmp
t_0 = ((y + x) / -y) * z
if (y < (-3.7429310762689856d+171)) then
tmp = t_0
else if (y < 3.5534662456086734d+168) then
tmp = (x + y) / (1.0d0 - (y / z))
else
tmp = t_0
end if
code = tmp
end function
public static double code(double x, double y, double z) {
double t_0 = ((y + x) / -y) * z;
double tmp;
if (y < -3.7429310762689856e+171) {
tmp = t_0;
} else if (y < 3.5534662456086734e+168) {
tmp = (x + y) / (1.0 - (y / z));
} else {
tmp = t_0;
}
return tmp;
}
def code(x, y, z): t_0 = ((y + x) / -y) * z tmp = 0 if y < -3.7429310762689856e+171: tmp = t_0 elif y < 3.5534662456086734e+168: tmp = (x + y) / (1.0 - (y / z)) else: tmp = t_0 return tmp
function code(x, y, z) t_0 = Float64(Float64(Float64(y + x) / Float64(-y)) * z) tmp = 0.0 if (y < -3.7429310762689856e+171) tmp = t_0; elseif (y < 3.5534662456086734e+168) tmp = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z))); else tmp = t_0; end return tmp end
function tmp_2 = code(x, y, z) t_0 = ((y + x) / -y) * z; tmp = 0.0; if (y < -3.7429310762689856e+171) tmp = t_0; elseif (y < 3.5534662456086734e+168) tmp = (x + y) / (1.0 - (y / z)); else tmp = t_0; end tmp_2 = tmp; end
code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y + x), $MachinePrecision] / (-y)), $MachinePrecision] * z), $MachinePrecision]}, If[Less[y, -3.7429310762689856e+171], t$95$0, If[Less[y, 3.5534662456086734e+168], N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]
\begin{array}{l}
\\
\begin{array}{l}
t_0 := \frac{y + x}{-y} \cdot z\\
\mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\
\;\;\;\;t\_0\\
\mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\
\;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\
\mathbf{else}:\\
\;\;\;\;t\_0\\
\end{array}
\end{array}
herbie shell --seed 2024215
(FPCore (x y z)
:name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
:precision binary64
:alt
(! :herbie-platform default (if (< y -3742931076268985600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (/ (+ y x) (- y)) z) (if (< y 3553466245608673400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (+ x y) (- 1 (/ y z))) (* (/ (+ y x) (- y)) z))))
(/ (+ x y) (- 1.0 (/ y z))))