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

Average Error: 1.5 → 0.8

Time: 10.5s

Precision: binary64

Cost: 968

Math

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

↓

\[\begin{array}{l} t_1 := x + y \cdot \frac{z - t}{z - a}\\ \mathbf{if}\;y \leq -2 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-185}:\\ \;\;\;\;x - \frac{y \cdot \left(t - z\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* y (/ (- z t) (- z a))))))
   (if (<= y -2e-112)
     t_1
     (if (<= y 5e-185) (- x (/ (* y (- t z)) (- z a))) t_1))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * ((z - t) / (z - a)));
	double tmp;
	if (y <= -2e-112) {
		tmp = t_1;
	} else if (y <= 5e-185) {
		tmp = x - ((y * (t - z)) / (z - a));
	} 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 - 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 + (y * ((z - t) / (z - a)))
    if (y <= (-2d-112)) then
        tmp = t_1
    else if (y <= 5d-185) then
        tmp = x - ((y * (t - z)) / (z - a))
    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 - a)));
}

↓

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * ((z - t) / (z - a)));
	double tmp;
	if (y <= -2e-112) {
		tmp = t_1;
	} else if (y <= 5e-185) {
		tmp = x - ((y * (t - z)) / (z - a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a):
	t_1 = x + (y * ((z - t) / (z - a)))
	tmp = 0
	if y <= -2e-112:
		tmp = t_1
	elif y <= 5e-185:
		tmp = x - ((y * (t - z)) / (z - a))
	else:
		tmp = t_1
	return tmp

Julia

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

↓

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(y * Float64(Float64(z - t) / Float64(z - a))))
	tmp = 0.0
	if (y <= -2e-112)
		tmp = t_1;
	elseif (y <= 5e-185)
		tmp = Float64(x - Float64(Float64(y * Float64(t - z)) / Float64(z - a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (y * ((z - t) / (z - a)));
	tmp = 0.0;
	if (y <= -2e-112)
		tmp = t_1;
	elseif (y <= 5e-185)
		tmp = x - ((y * (t - z)) / (z - a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2e-112], t$95$1, If[LessEqual[y, 5e-185], N[(x - N[(N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Target

Original	1.5
Target	1.4
Herbie	0.8

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

Derivation

1. Split input into 2 regimes

2. if y < -1.9999999999999999e-112 or 5.0000000000000003e-185 < y

   1. Initial program 1.0

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

   if -1.9999999999999999e-112 < y < 5.0000000000000003e-185

   1. Initial program 2.9

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

   2. Simplified 0.3

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(z - t\right)}{z - a}} \]
      Proof
      (+.f64 x (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a))): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite<= associate-*r/_binary64 (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))))): 18 points increase in error, 59 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-112}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-185}:\\ \;\;\;\;x - \frac{y \cdot \left(t - z\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{z - a}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.6
Cost	3796

\[\begin{array}{l} t_1 := x + \frac{y}{1 - \frac{a}{z}}\\ t_2 := y \cdot \frac{z - t}{z - a}\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{+121}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{+47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -20:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq -5 \cdot 10^{-19}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;t_2 \leq 10^{+87}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	12.7
Cost	2124

\[\begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+75}:\\ \;\;\;\;x + \frac{y}{\frac{-z}{t}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{elif}\;t_1 \leq 1.000002:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot t_1\\ \end{array} \]

Alternative 3
Error	9.1
Cost	2124

\[\begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+75}:\\ \;\;\;\;x + \frac{y}{\frac{-z}{t}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+49}:\\ \;\;\;\;x + \frac{y}{\frac{z}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \end{array} \]

Alternative 4
Error	16.8
Cost	976

\[\begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{-31}:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{elif}\;a \leq -6.8 \cdot 10^{-259}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{-270}:\\ \;\;\;\;x + \frac{y}{\frac{-z}{t}}\\ \mathbf{elif}\;a \leq 5.4 \cdot 10^{+39}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \end{array} \]

Alternative 5
Error	1.6
Cost	968

\[\begin{array}{l} t_1 := x + y \cdot \frac{z - t}{z - a}\\ \mathbf{if}\;z \leq -1.35 \cdot 10^{-213}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-268}:\\ \;\;\;\;x + \frac{y}{a} \cdot \left(t - z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	16.2
Cost	712

\[\begin{array}{l} t_1 := x + y \cdot \frac{t}{a}\\ \mathbf{if}\;a \leq -1.3 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+29}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	14.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+16}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 64:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 8
Error	14.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+16}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 7.7:\\ \;\;\;\;x + t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]

Alternative 9
Error	20.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{+142}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{+173}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	27.3
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-100}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-128}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	28.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022329 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, A"
  :precision binary64

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

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