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

Average Error: 10.6 → 6.0

Time: 8.0s

Precision: binary64

Cost: 1992

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{t - z}{a} \cdot y + x\\ \mathbf{elif}\;t_1 \leq 10^{+264}:\\ \;\;\;\;x + t_1\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

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 (/ (* y (- z t)) (- z a))))
   (if (<= t_1 (- INFINITY))
     (+ (* (/ (- t z) a) y) x)
     (if (<= t_1 1e+264) (+ x t_1) (+ x y)))))

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 = (y * (z - t)) / (z - a);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (((t - z) / a) * y) + x;
	} else if (t_1 <= 1e+264) {
		tmp = x + t_1;
	} else {
		tmp = x + y;
	}
	return tmp;
}

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 = (y * (z - t)) / (z - a);
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (((t - z) / a) * y) + x;
	} else if (t_1 <= 1e+264) {
		tmp = x + t_1;
	} else {
		tmp = x + y;
	}
	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 = (y * (z - t)) / (z - a)
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (((t - z) / a) * y) + x
	elif t_1 <= 1e+264:
		tmp = x + t_1
	else:
		tmp = x + y
	return tmp

Julia

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

↓

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(z - t)) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(t - z) / a) * y) + x);
	elseif (t_1 <= 1e+264)
		tmp = Float64(x + t_1);
	else
		tmp = Float64(x + y);
	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 = (y * (z - t)) / (z - a);
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (((t - z) / a) * y) + x;
	elseif (t_1 <= 1e+264)
		tmp = x + t_1;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+264], N[(x + t$95$1), $MachinePrecision], N[(x + y), $MachinePrecision]]]]

Target

Original	10.6
Target	1.2
Herbie	6.0

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < -inf.0

   1. Initial program 64.0

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

   2. Taylor expanded in a around inf 64.0

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

   3. Simplified 64.0

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

      [Start] 64.0	\[ x + -1 \cdot \frac{\left(z - t\right) \cdot y}{a} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 64.0	\[ x + \color{blue}{\frac{\left(z - t\right) \cdot y}{a} \cdot -1} \]
      rational_best_oopsla_all_46_json_45_simplify-92 [=>] 64.0	\[ x + \color{blue}{\left(-\frac{\left(z - t\right) \cdot y}{a}\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 64.0	\[ x + \left(-\frac{\color{blue}{y \cdot \left(z - t\right)}}{a}\right) \]

   4. Taylor expanded in y around 0 43.2

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

   5. Taylor expanded in a around 0 43.2

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

   if -inf.0 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a)) < 1.00000000000000004e264

   1. Initial program 0.2

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

   if 1.00000000000000004e264 < (/.f64 (*.f64 y (-.f64 z t)) (-.f64 z a))

   1. Initial program 57.4

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

   2. Taylor expanded in z around inf 25.5

      \[\leadsto x + \color{blue}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 6.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{z - a} \leq -\infty:\\ \;\;\;\;\frac{t - z}{a} \cdot y + x\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{z - a} \leq 10^{+264}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternatives

Alternative 1
Error	15.7
Cost	1236

\[\begin{array}{l} t_1 := \frac{t - z}{a} \cdot y + x\\ \mathbf{if}\;a \leq -60000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{-182}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z}\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{-37}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{elif}\;a \leq 9 \cdot 10^{+14}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 7.7 \cdot 10^{+93}:\\ \;\;\;\;\frac{t}{a} \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	11.7
Cost	1168

\[\begin{array}{l} t_1 := x + \frac{y \cdot \left(-t\right)}{z - a}\\ \mathbf{if}\;z \leq -9 \cdot 10^{+121}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 1720000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+48}:\\ \;\;\;\;x + \frac{y \cdot z}{z - a}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+104}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 3
Error	14.3
Cost	972

\[\begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+50}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-28}:\\ \;\;\;\;\frac{t}{a} \cdot y + x\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+122}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 4
Error	15.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+43}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 2.46 \cdot 10^{-23}:\\ \;\;\;\;x + \frac{y \cdot t}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 5
Error	14.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -3.25 \cdot 10^{+50}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-26}:\\ \;\;\;\;\frac{t}{a} \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]

Alternative 6
Error	19.5
Cost	456

\[\begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{+156}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{+193}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	28.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

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

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