Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3

Average Error: 2.1 → 12.3

Time: 6.5s

Precision: binary64

Cost: 968

Math

\[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

↓

\[\begin{array}{l} t_1 := x + a \cdot \frac{z}{\left(1 + t\right) - z}\\ \mathbf{if}\;z \leq -1.8217880177163876 \cdot 10^{-65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.7015733050017065 \cdot 10^{-175}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* a (/ z (- (+ 1.0 t) z))))))
   (if (<= z -1.8217880177163876e-65)
     t_1
     (if (<= z 2.7015733050017065e-175) (- x (* a y)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / a));
}

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (a * (z / ((1.0 + t) - z)));
	double tmp;
	if (z <= -1.8217880177163876e-65) {
		tmp = t_1;
	} else if (z <= 2.7015733050017065e-175) {
		tmp = x - (a * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x - ((y - z) / (((t - z) + 1.0d0) / a))
end function

↓

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (a * (z / ((1.0d0 + t) - z)))
    if (z <= (-1.8217880177163876d-65)) then
        tmp = t_1
    else if (z <= 2.7015733050017065d-175) then
        tmp = x - (a * y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / a));
}

↓

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (a * (z / ((1.0 + t) - z)));
	double tmp;
	if (z <= -1.8217880177163876e-65) {
		tmp = t_1;
	} else if (z <= 2.7015733050017065e-175) {
		tmp = x - (a * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	return x - ((y - z) / (((t - z) + 1.0) / a))

↓

def code(x, y, z, t, a):
	t_1 = x + (a * (z / ((1.0 + t) - z)))
	tmp = 0
	if z <= -1.8217880177163876e-65:
		tmp = t_1
	elif z <= 2.7015733050017065e-175:
		tmp = x - (a * y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	return Float64(x - Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a)))
end

↓

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(a * Float64(z / Float64(Float64(1.0 + t) - z))))
	tmp = 0.0
	if (z <= -1.8217880177163876e-65)
		tmp = t_1;
	elseif (z <= 2.7015733050017065e-175)
		tmp = Float64(x - Float64(a * y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x - ((y - z) / (((t - z) + 1.0) / a));
end

↓

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (a * (z / ((1.0 + t) - z)));
	tmp = 0.0;
	if (z <= -1.8217880177163876e-65)
		tmp = t_1;
	elseif (z <= 2.7015733050017065e-175)
		tmp = x - (a * y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x - N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(a * N[(z / N[(N[(1.0 + t), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.8217880177163876e-65], t$95$1, If[LessEqual[z, 2.7015733050017065e-175], N[(x - N[(a * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Target

Original	2.1
Target	0.3
Herbie	12.3

\[x - \frac{y - z}{\left(t - z\right) + 1} \cdot a \]

Derivation

1. Split input into 2 regimes

2. if z < -1.8217880177163876e-65 or 2.7015733050017065e-175 < z

   1. Initial program 2.8

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Simplified 2.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a}{t - \left(z + -1\right)}, z - y, x\right)} \]

   3. Taylor expanded in y around 0 20.2

      \[\leadsto \color{blue}{\frac{a \cdot z}{\left(1 + t\right) - z} + x} \]

   4. Simplified 10.4

      \[\leadsto \color{blue}{x + a \cdot \frac{z}{\left(1 + t\right) - z}} \]

   if -1.8217880177163876e-65 < z < 2.7015733050017065e-175

   1. Initial program 0.5

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

   2. Taylor expanded in z around 0 5.5

      \[\leadsto x - \color{blue}{\frac{a \cdot y}{1 + t}} \]

   3. Taylor expanded in t around 0 17.1

      \[\leadsto x - \color{blue}{y \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 12.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8217880177163876 \cdot 10^{-65}:\\ \;\;\;\;x + a \cdot \frac{z}{\left(1 + t\right) - z}\\ \mathbf{elif}\;z \leq 2.7015733050017065 \cdot 10^{-175}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \frac{z}{\left(1 + t\right) - z}\\ \end{array} \]

Alternatives

Alternative 1
Error	17.9
Cost	1104

\[\begin{array}{l} t_1 := x + z \cdot \frac{a}{1 - z}\\ t_2 := x - \frac{y}{\frac{t}{a}}\\ \mathbf{if}\;t \leq -338435360563229300:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 5.29982329399835 \cdot 10^{-241}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.040065785782773 \cdot 10^{-185}:\\ \;\;\;\;y \cdot \frac{-a}{1 - z}\\ \mathbf{elif}\;t \leq 1.2184983543792464 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	16.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -7.584525728927753 \cdot 10^{-7}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 1.1325324260397663 \cdot 10^{-44}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 3
Error	18.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -1.7088307887448715 \cdot 10^{-16}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 2.4221986818390974 \cdot 10^{-18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 4
Error	26.0
Cost	392

\[\begin{array}{l} \mathbf{if}\;a \leq -2.7306642168438476 \cdot 10^{+154}:\\ \;\;\;\;-a\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{+187}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-a\\ \end{array} \]

Alternative 5
Error	27.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022228 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (- x (* (/ (- y z) (+ (- t z) 1.0)) a))

  (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))