Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A

Average Error: 10.9 → 1.4

Time: 8.8s

Precision: binary64

Cost: 704

Math

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

↓

\[x + \frac{y - z}{a - z} \cdot t \]

FPCore

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

↓

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

C

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

↓

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

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) / (a - z))
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) / (a - z)) * t)
end function

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	10.9
Target	0.5
Herbie	1.4

\[\begin{array}{l} \mathbf{if}\;t < -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \mathbf{elif}\;t < 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a - z} \cdot t\\ \end{array} \]

Derivation

1. Initial program 10.9

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

2. Simplified 1.4

   \[\leadsto \color{blue}{x + \frac{y - z}{a - z} \cdot t} \]
   Proof
   (+.f64 x (*.f64 (/.f64 (-.f64 y z) (-.f64 a z)) t)): 0 points increase in error, 0 points decrease in error
   (+.f64 x (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)))): 47 points increase in error, 14 points decrease in error

3. Final simplification 1.4

   \[\leadsto x + \frac{y - z}{a - z} \cdot t \]

Alternatives

Alternative 1
Error	15.0
Cost	976

\[\begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+14}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq -5.1 \cdot 10^{-70}:\\ \;\;\;\;x - z \cdot \frac{t}{a}\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-136}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-15}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \]

Alternative 2
Error	15.0
Cost	976

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+20}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-76}:\\ \;\;\;\;x - t \cdot \frac{z}{a}\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{-137}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-17}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \]

Alternative 3
Error	15.0
Cost	976

\[\begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+15}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{-78}:\\ \;\;\;\;x - t \cdot \frac{z}{a}\\ \mathbf{elif}\;z \leq -9.8 \cdot 10^{-142}:\\ \;\;\;\;x - \frac{y \cdot t}{z}\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-17}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \]

Alternative 4
Error	10.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+122}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+39}:\\ \;\;\;\;x + t \cdot \frac{y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \]

Alternative 5
Error	8.6
Cost	840

\[\begin{array}{l} t_1 := x + t \cdot \frac{y}{a - z}\\ \mathbf{if}\;y \leq -3.9 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+21}:\\ \;\;\;\;x + t \cdot \frac{z}{z - a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	8.3
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+50}:\\ \;\;\;\;x + \frac{y}{\frac{a - z}{t}}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+18}:\\ \;\;\;\;x + t \cdot \frac{z}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + t \cdot \frac{y}{a - z}\\ \end{array} \]

Alternative 7
Error	15.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-137}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-16}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \]

Alternative 8
Error	15.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-137}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-17}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \]

Alternative 9
Error	20.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-147}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-108}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \]

Alternative 10
Error	29.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022325 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (if (< t -1.0682974490174067e-39) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t))))

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