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

Average Error: 2.0 → 0.2

Time: 19.4s

Precision: binary64

Cost: 1216

Math

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

↓

\[\begin{array}{l} t_1 := \left(1 + t\right) - z\\ a \cdot \left(\frac{z}{t_1} - \frac{y}{t_1}\right) + x \end{array} \]

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 (- (+ 1.0 t) z))) (+ (* a (- (/ z t_1) (/ y t_1))) x)))

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 = (1.0 + t) - z;
	return (a * ((z / t_1) - (y / t_1))) + x;
}

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
    t_1 = (1.0d0 + t) - z
    code = (a * ((z / t_1) - (y / t_1))) + x
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 = (1.0 + t) - z;
	return (a * ((z / t_1) - (y / t_1))) + x;
}

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 = (1.0 + t) - z
	return (a * ((z / t_1) - (y / t_1))) + x

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(Float64(1.0 + t) - z)
	return Float64(Float64(a * Float64(Float64(z / t_1) - Float64(y / t_1))) + x)
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)
	t_1 = (1.0 + t) - z;
	tmp = (a * ((z / t_1) - (y / t_1))) + x;
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[(N[(1.0 + t), $MachinePrecision] - z), $MachinePrecision]}, N[(N[(a * N[(N[(z / t$95$1), $MachinePrecision] - N[(y / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Target

Original	2.0
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 2.0

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

2. Taylor expanded in a around 0 0.2

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

3. Final simplification 0.2

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

Alternatives

Alternative 1
Error	1.0
Cost	2376

\[\begin{array}{l} t_1 := \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\\ t_2 := x - t_1\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-170}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{a \cdot z}{\left(1 + t\right) - z} + x\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	6.9
Cost	1236

\[\begin{array}{l} t_1 := a \cdot \frac{z - y}{t} + x\\ \mathbf{if}\;t \leq -4.4 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+78}:\\ \;\;\;\;x - \frac{y - z}{\frac{1 - z}{a}}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+171}:\\ \;\;\;\;a \cdot \left(-\frac{y}{\left(1 + t\right) - z}\right) + x\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{+184}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+269}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - z}{\frac{t}{a}}\\ \end{array} \]

Alternative 3
Error	11.4
Cost	1232

\[\begin{array}{l} t_1 := x - \frac{a \cdot \left(y - z\right)}{1 - z}\\ \mathbf{if}\;t \leq -5 \cdot 10^{+14}:\\ \;\;\;\;a \cdot \frac{z - y}{t} + x\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-265}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{-256}:\\ \;\;\;\;x - \frac{y - z}{-\frac{z}{a}}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+78}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - z}{\frac{t}{a}}\\ \end{array} \]

Alternative 4
Error	19.9
Cost	1108

\[\begin{array}{l} t_1 := a \cdot \left(-\frac{y}{1 + t}\right)\\ \mathbf{if}\;z \leq -5 \cdot 10^{-7}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{-46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.5 \cdot 10^{-60}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -7.3 \cdot 10^{-68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+18}:\\ \;\;\;\;x - \frac{y \cdot a}{t}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 5
Error	6.5
Cost	968

\[\begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{+15}:\\ \;\;\;\;a \cdot \frac{z - y}{t} + x\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+78}:\\ \;\;\;\;x - \frac{y - z}{\frac{1 - z}{a}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y - z}{\frac{t}{a}}\\ \end{array} \]

Alternative 6
Error	10.2
Cost	904

\[\begin{array}{l} t_1 := x - \frac{y - z}{-\frac{z}{a}}\\ \mathbf{if}\;z \leq -1.32 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+17}:\\ \;\;\;\;x - \frac{y \cdot a}{1 + t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	16.3
Cost	840

\[\begin{array}{l} t_1 := a \cdot \frac{z - y}{t} + x\\ \mathbf{if}\;t \leq -3.3 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+88}:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	11.1
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+33}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{+19}:\\ \;\;\;\;x - \frac{y \cdot a}{1 + t}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 9
Error	10.8
Cost	840

\[\begin{array}{l} t_1 := \left(\frac{y \cdot a}{z} + x\right) - a\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+17}:\\ \;\;\;\;x - \frac{y \cdot a}{1 + t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	18.1
Cost	776

\[\begin{array}{l} t_1 := a \cdot \left(-\frac{y}{t}\right) + x\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+89}:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	20.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-47}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{a \cdot z}{t} + x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 12
Error	19.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -2.75 \cdot 10^{+32}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{+17}:\\ \;\;\;\;x - \frac{y \cdot a}{t}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 13
Error	19.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -52000:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+14}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 14
Error	27.1
Cost	392

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-271}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-252}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Error	27.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(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))))