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

Average Error: 2.2 → 0.2

Time: 14.3s

Precision: binary64

Cost: 832

Math

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

↓

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

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

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) {
	return x - (((y - z) / ((t - z) + 1.0)) * a);
}

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
    code = x - (((y - z) / ((t - z) + 1.0d0)) * a)
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) {
	return x - (((y - z) / ((t - z) + 1.0)) * a);
}

Python

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

↓

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

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)
	return Float64(x - Float64(Float64(Float64(y - z) / Float64(Float64(t - z) + 1.0)) * a))
end

MATLAB

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

↓

function tmp = code(x, y, z, t, a)
	tmp = x - (((y - z) / ((t - z) + 1.0)) * a);
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_] := N[(x - N[(N[(N[(y - z), $MachinePrecision] / N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]

Target

Original	2.2
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 2.2

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

2. Simplified 0.2

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

   [Start] 2.2	\[ x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   associate-/r/ [=>] 0.2	\[ x - \color{blue}{\frac{y - z}{\left(t - z\right) + 1} \cdot a} \]

3. Final simplification 0.2

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

Alternatives

Alternative 1
Error	14.9
Cost	1500

\[\begin{array}{l} t_1 := x + \frac{a}{t} \cdot \left(z - y\right)\\ t_2 := x + \left(y \cdot \frac{a}{z} - a\right)\\ \mathbf{if}\;z \leq -1.92 \cdot 10^{+65}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.06 \cdot 10^{-94}:\\ \;\;\;\;x + z \cdot a\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-239}:\\ \;\;\;\;x - \frac{a}{\frac{t}{y}}\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-164}:\\ \;\;\;\;x - y \cdot a\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	16.2
Cost	1369

\[\begin{array}{l} t_1 := x + \frac{a}{t} \cdot \left(z - y\right)\\ \mathbf{if}\;t \leq -56000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-215}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-169}:\\ \;\;\;\;x - y \cdot a\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+45} \lor \neg \left(t \leq 1.42 \cdot 10^{+76}\right) \land t \leq 6.4 \cdot 10^{+96}:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	15.2
Cost	1368

\[\begin{array}{l} t_1 := x + \left(y \cdot \frac{a}{z} - a\right)\\ t_2 := x - \frac{a}{\frac{t}{y - z}}\\ \mathbf{if}\;z \leq -1.85 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{+47}:\\ \;\;\;\;x + \frac{a}{t} \cdot \left(z - y\right)\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-248}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{-163}:\\ \;\;\;\;x - y \cdot a\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+33}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	8.7
Cost	1168

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+65}:\\ \;\;\;\;x + \left(y \cdot \frac{a}{z} - a\right)\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{+47}:\\ \;\;\;\;x + \frac{a}{t} \cdot \left(z - y\right)\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-40}:\\ \;\;\;\;x - \frac{z \cdot a}{z - 1}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+32}:\\ \;\;\;\;x - \frac{a}{\frac{t + 1}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - y}{\frac{-z}{a}}\\ \end{array} \]

Alternative 5
Error	17.1
Cost	1108

\[\begin{array}{l} t_1 := x - y \cdot a\\ t_2 := x - a \cdot \frac{y}{t}\\ \mathbf{if}\;z \leq -4 \cdot 10^{-93}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -8.2 \cdot 10^{-246}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-71}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 54000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x - \left(a + \frac{a}{z}\right)\\ \end{array} \]

Alternative 6
Error	17.0
Cost	1108

\[\begin{array}{l} t_1 := x - y \cdot a\\ \mathbf{if}\;z \leq -2.7 \cdot 10^{-89}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -8.8 \cdot 10^{-237}:\\ \;\;\;\;x - \frac{a}{\frac{t}{y}}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-72}:\\ \;\;\;\;x - a \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 54000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x - \left(a + \frac{a}{z}\right)\\ \end{array} \]

Alternative 7
Error	8.2
Cost	1105

\[\begin{array}{l} t_1 := x + \left(y \cdot \frac{a}{z} - a\right)\\ \mathbf{if}\;z \leq -2 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6 \cdot 10^{+47}:\\ \;\;\;\;x + \frac{a}{t} \cdot \left(z - y\right)\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-40} \lor \neg \left(z \leq 6.4 \cdot 10^{+32}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x - \frac{a}{\frac{t + 1}{y}}\\ \end{array} \]

Alternative 8
Error	8.1
Cost	1104

\[\begin{array}{l} t_1 := x + \left(y \cdot \frac{a}{z} - a\right)\\ \mathbf{if}\;z \leq -4.7 \cdot 10^{+66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{+47}:\\ \;\;\;\;x + \frac{a}{t} \cdot \left(z - y\right)\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-40}:\\ \;\;\;\;x - \frac{a}{\frac{z - 1}{z}}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+32}:\\ \;\;\;\;x - \frac{a}{\frac{t + 1}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	8.1
Cost	1104

\[\begin{array}{l} t_1 := x + \left(y \cdot \frac{a}{z} - a\right)\\ \mathbf{if}\;z \leq -2.05 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -9.2 \cdot 10^{+47}:\\ \;\;\;\;x + \frac{a}{t} \cdot \left(z - y\right)\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-43}:\\ \;\;\;\;x - \frac{z \cdot a}{z - 1}\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+32}:\\ \;\;\;\;x - \frac{a}{\frac{t + 1}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	6.4
Cost	1100

\[\begin{array}{l} t_1 := x - a \cdot \frac{z}{z + \left(-1 - t\right)}\\ \mathbf{if}\;t \leq -4.2 \cdot 10^{+46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -56000000:\\ \;\;\;\;x + \frac{a}{t} \cdot \left(z - y\right)\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{+20}:\\ \;\;\;\;x - \frac{a}{\frac{1 - z}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	17.1
Cost	980

\[\begin{array}{l} t_1 := x - y \cdot a\\ t_2 := x - a \cdot \frac{y}{t}\\ \mathbf{if}\;z \leq -3 \cdot 10^{-89}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -6.7 \cdot 10^{-239}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.65 \cdot 10^{-163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-69}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 430000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 12
Error	17.9
Cost	976

\[\begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-89}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-241}:\\ \;\;\;\;x - \frac{a}{\frac{t}{y}}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-176}:\\ \;\;\;\;x - y \cdot a\\ \mathbf{elif}\;z \leq 130000000:\\ \;\;\;\;x + \frac{a}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \left(a + \frac{a}{z}\right)\\ \end{array} \]

Alternative 13
Error	7.3
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -1.01 \cdot 10^{-56} \lor \neg \left(z \leq 1.8 \cdot 10^{-131}\right):\\ \;\;\;\;x - a \cdot \frac{z}{z + \left(-1 - t\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{a}{\frac{t + 1}{y}}\\ \end{array} \]

Alternative 14
Error	17.2
Cost	852

\[\begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{-89}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -2.8 \cdot 10^{-208}:\\ \;\;\;\;x - y \cdot \frac{a}{t}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-164}:\\ \;\;\;\;x - y \cdot a\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-91}:\\ \;\;\;\;x + z \cdot a\\ \mathbf{elif}\;z \leq 2200000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 15
Error	16.7
Cost	720

\[\begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-40}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-164}:\\ \;\;\;\;x - y \cdot a\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-92}:\\ \;\;\;\;x + z \cdot a\\ \mathbf{elif}\;z \leq 100000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 16
Error	16.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-40}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 90000000:\\ \;\;\;\;x - y \cdot a\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 17
Error	18.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-40}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 2150000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 18
Error	26.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023016 
(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))))