Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3

Average Error: 25.2 → 8.9

Time: 17.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \left(\frac{z}{a - t} - \frac{t}{a - t}\right) \cdot y\\ t_2 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -3.478895894122768 \cdot 10^{-307}:\\ \;\;\;\;\left(\frac{t \cdot x}{a - t} + \left(\frac{y \cdot z}{a - t} + x\right)\right) - \left(\frac{z \cdot x}{a - t} + \frac{y \cdot t}{a - t}\right)\\ \mathbf{elif}\;t_2 \leq 8.768236575039256 \cdot 10^{-293}:\\ \;\;\;\;\left(\frac{y \cdot a}{t} + \left(y + \frac{z \cdot x}{t}\right)\right) - \left(\frac{a \cdot x}{t} + \frac{y \cdot z}{t}\right)\\ \mathbf{elif}\;t_2 \leq 2.0104648879690074 \cdot 10^{+302}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- (/ z (- a t)) (/ t (- a t))) y))
        (t_2 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -3.478895894122768e-307)
       (-
        (+ (/ (* t x) (- a t)) (+ (/ (* y z) (- a t)) x))
        (+ (/ (* z x) (- a t)) (/ (* y t) (- a t))))
       (if (<= t_2 8.768236575039256e-293)
         (-
          (+ (/ (* y a) t) (+ y (/ (* z x) t)))
          (+ (/ (* a x) t) (/ (* y z) t)))
         (if (<= t_2 2.0104648879690074e+302) t_2 t_1))))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z / (a - t)) - (t / (a - t))) * y;
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -3.478895894122768e-307) {
		tmp = (((t * x) / (a - t)) + (((y * z) / (a - t)) + x)) - (((z * x) / (a - t)) + ((y * t) / (a - t)));
	} else if (t_2 <= 8.768236575039256e-293) {
		tmp = (((y * a) / t) + (y + ((z * x) / t))) - (((a * x) / t) + ((y * z) / t));
	} else if (t_2 <= 2.0104648879690074e+302) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

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

↓

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z / (a - t)) - (t / (a - t))) * y;
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= -3.478895894122768e-307) {
		tmp = (((t * x) / (a - t)) + (((y * z) / (a - t)) + x)) - (((z * x) / (a - t)) + ((y * t) / (a - t)));
	} else if (t_2 <= 8.768236575039256e-293) {
		tmp = (((y * a) / t) + (y + ((z * x) / t))) - (((a * x) / t) + ((y * z) / t));
	} else if (t_2 <= 2.0104648879690074e+302) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a):
	t_1 = ((z / (a - t)) - (t / (a - t))) * y
	t_2 = x + (((y - x) * (z - t)) / (a - t))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= -3.478895894122768e-307:
		tmp = (((t * x) / (a - t)) + (((y * z) / (a - t)) + x)) - (((z * x) / (a - t)) + ((y * t) / (a - t)))
	elif t_2 <= 8.768236575039256e-293:
		tmp = (((y * a) / t) + (y + ((z * x) / t))) - (((a * x) / t) + ((y * z) / t))
	elif t_2 <= 2.0104648879690074e+302:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

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

↓

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z / Float64(a - t)) - Float64(t / Float64(a - t))) * y)
	t_2 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -3.478895894122768e-307)
		tmp = Float64(Float64(Float64(Float64(t * x) / Float64(a - t)) + Float64(Float64(Float64(y * z) / Float64(a - t)) + x)) - Float64(Float64(Float64(z * x) / Float64(a - t)) + Float64(Float64(y * t) / Float64(a - t))));
	elseif (t_2 <= 8.768236575039256e-293)
		tmp = Float64(Float64(Float64(Float64(y * a) / t) + Float64(y + Float64(Float64(z * x) / t))) - Float64(Float64(Float64(a * x) / t) + Float64(Float64(y * z) / t)));
	elseif (t_2 <= 2.0104648879690074e+302)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((z / (a - t)) - (t / (a - t))) * y;
	t_2 = x + (((y - x) * (z - t)) / (a - t));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= -3.478895894122768e-307)
		tmp = (((t * x) / (a - t)) + (((y * z) / (a - t)) + x)) - (((z * x) / (a - t)) + ((y * t) / (a - t)));
	elseif (t_2 <= 8.768236575039256e-293)
		tmp = (((y * a) / t) + (y + ((z * x) / t))) - (((a * x) / t) + ((y * z) / t));
	elseif (t_2 <= 2.0104648879690074e+302)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] - N[(t / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -3.478895894122768e-307], N[(N[(N[(N[(t * x), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(y * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(z * x), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + N[(N[(y * t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 8.768236575039256e-293], N[(N[(N[(N[(y * a), $MachinePrecision] / t), $MachinePrecision] + N[(y + N[(N[(z * x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a * x), $MachinePrecision] / t), $MachinePrecision] + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0104648879690074e+302], t$95$2, t$95$1]]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	25.2
Target	9.6
Herbie	8.9

\[\begin{array}{l} \mathbf{if}\;a < -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a < 3.774403170083174 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -inf.0 or 2.01046488796900736e302 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 63.7

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

   2. Taylor expanded in y around inf 25.5

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

   if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -3.478895894122768e-307

   1. Initial program 2.6

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

   2. Taylor expanded in x around 0 2.2

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

   if -3.478895894122768e-307 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 8.76824e-293

   1. Initial program 59.9

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

   2. Taylor expanded in t around inf 1.6

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

   if 8.76824e-293 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 2.01046488796900736e302

   1. Initial program 2.1

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
3. Recombined 4 regimes into one program.

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -\infty:\\ \;\;\;\;\left(\frac{z}{a - t} - \frac{t}{a - t}\right) \cdot y\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -3.478895894122768 \cdot 10^{-307}:\\ \;\;\;\;\left(\frac{t \cdot x}{a - t} + \left(\frac{y \cdot z}{a - t} + x\right)\right) - \left(\frac{z \cdot x}{a - t} + \frac{y \cdot t}{a - t}\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 8.768236575039256 \cdot 10^{-293}:\\ \;\;\;\;\left(\frac{y \cdot a}{t} + \left(y + \frac{z \cdot x}{t}\right)\right) - \left(\frac{a \cdot x}{t} + \frac{y \cdot z}{t}\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 2.0104648879690074 \cdot 10^{+302}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{a - t} - \frac{t}{a - t}\right) \cdot y\\ \end{array} \]

Reproduce

herbie shell --seed 2022152 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< a -1.6153062845442575e-142) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))

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