Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B

Average Error: 1.4 → 0.4

Time: 11.5s

Precision: binary64

Cost: 969

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-79} \lor \neg \left(y \leq 2 \cdot 10^{-29}\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= y -1e-79) (not (<= y 2e-29)))
   (+ x (* y (/ (- z t) (- a t))))
   (+ x (/ (* y (- z t)) (- a t)))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y <= -1e-79) || !(y <= 2e-29)) {
		tmp = x + (y * ((z - t) / (a - t)));
	} else {
		tmp = x + ((y * (z - t)) / (a - t));
	}
	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) / (a - t)))
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) :: tmp
    if ((y <= (-1d-79)) .or. (.not. (y <= 2d-29))) then
        tmp = x + (y * ((z - t) / (a - t)))
    else
        tmp = x + ((y * (z - t)) / (a - t))
    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) / (a - t)));
}

↓

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y <= -1e-79) || !(y <= 2e-29)) {
		tmp = x + (y * ((z - t) / (a - t)));
	} else {
		tmp = x + ((y * (z - t)) / (a - t));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a):
	tmp = 0
	if (y <= -1e-79) or not (y <= 2e-29):
		tmp = x + (y * ((z - t) / (a - t)))
	else:
		tmp = x + ((y * (z - t)) / (a - t))
	return tmp

Julia

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

↓

function code(x, y, z, t, a)
	tmp = 0.0
	if ((y <= -1e-79) || !(y <= 2e-29))
		tmp = Float64(x + Float64(y * Float64(Float64(z - t) / Float64(a - t))));
	else
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) / Float64(a - t)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((y <= -1e-79) || ~((y <= 2e-29)))
		tmp = x + (y * ((z - t) / (a - t)));
	else
		tmp = x + ((y * (z - t)) / (a - t));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[y, -1e-79], N[Not[LessEqual[y, 2e-29]], $MachinePrecision]], N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	1.4
Target	0.4
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;y < -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if y < -1e-79 or 1.99999999999999989e-29 < y

   1. Initial program 0.6

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

   if -1e-79 < y < 1.99999999999999989e-29

   1. Initial program 2.5

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

   2. Simplified 0.2

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(z - t\right)}{a - t}} \]
      Proof
      (+.f64 x (/.f64 (*.f64 y (-.f64 z t)) (-.f64 a t))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite<= associate-*r/_binary64 (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t))))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-79} \lor \neg \left(y \leq 2 \cdot 10^{-29}\right):\\ \;\;\;\;x + y \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a - t}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.0
Cost	1480

\[\begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t_1 \leq 0.02:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \mathbf{elif}\;t_1 \leq 400:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \end{array} \]

Alternative 2
Error	21.5
Cost	844

\[\begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{-66}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-220}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-150}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 3
Error	12.9
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-58} \lor \neg \left(x \leq 6 \cdot 10^{-120}\right):\\ \;\;\;\;x - t \cdot \frac{y}{a - t}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \end{array} \]

Alternative 4
Error	12.0
Cost	840

\[\begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+47}:\\ \;\;\;\;x + \frac{y}{\frac{t}{t - z}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-91}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x - t \cdot \frac{y}{a - t}\\ \end{array} \]

Alternative 5
Error	14.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+44}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-85}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 6
Error	1.4
Cost	704

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

Alternative 7
Error	20.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{-66}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-158}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 8
Error	28.6
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-48}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-135}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	29.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022343 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< y -8.508084860551241e-17) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2.894426862792089e-49) (+ x (* (* y (- z t)) (/ 1.0 (- a t)))) (+ x (* y (/ (- z t) (- a t))))))

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