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

Average Error: 24.4 → 7.1

Time: 8.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t_1 \leq -2.697284317091032 \cdot 10^{-251}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\left(y + \left(\frac{x \cdot z}{t} + \frac{y \cdot a}{t}\right)\right) - \left(\frac{x \cdot a}{t} + \frac{y \cdot z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{1}{\frac{a - t}{z - t}}, x\right)\\ \end{array} \]

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 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (<= t_1 -2.697284317091032e-251)
     (fma (- y x) (/ (- z t) (- a t)) x)
     (if (<= t_1 0.0)
       (-
        (+ y (+ (/ (* x z) t) (/ (* y a) t)))
        (+ (/ (* x a) t) (/ (* y z) t)))
       (fma (- y x) (/ 1.0 (/ (- a t) (- z t))) x)))))

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 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_1 <= -2.697284317091032e-251) {
		tmp = fma((y - x), ((z - t) / (a - t)), x);
	} else if (t_1 <= 0.0) {
		tmp = (y + (((x * z) / t) + ((y * a) / t))) - (((x * a) / t) + ((y * z) / t));
	} else {
		tmp = fma((y - x), (1.0 / ((a - t) / (z - t))), x);
	}
	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(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if (t_1 <= -2.697284317091032e-251)
		tmp = fma(Float64(y - x), Float64(Float64(z - t) / Float64(a - t)), x);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(y + Float64(Float64(Float64(x * z) / t) + Float64(Float64(y * a) / t))) - Float64(Float64(Float64(x * a) / t) + Float64(Float64(y * z) / t)));
	else
		tmp = fma(Float64(y - x), Float64(1.0 / Float64(Float64(a - t) / Float64(z - t))), x);
	end
	return 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[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2.697284317091032e-251], N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(y + N[(N[(N[(x * z), $MachinePrecision] / t), $MachinePrecision] + N[(N[(y * a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(x * a), $MachinePrecision] / t), $MachinePrecision] + N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * N[(1.0 / N[(N[(a - t), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.4
Target	9.1
Herbie	7.1

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -2.69728431709103211e-251

   1. Initial program 21.1

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

   2. Simplified 6.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Applied div-inv_binary64 6.8

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

   4. Applied associate-*r/_binary64 6.7

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

   if -2.69728431709103211e-251 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 55.7

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

   2. Simplified 55.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Taylor expanded in t around inf 5.2

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

   if 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 21.5

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

   2. Simplified 7.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)} \]

   3. Applied clear-num_binary64 7.8

      \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{1}{\frac{a - t}{z - t}}}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 7.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -2.697284317091032 \cdot 10^{-251}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a - t}, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 0:\\ \;\;\;\;\left(y + \left(\frac{x \cdot z}{t} + \frac{y \cdot a}{t}\right)\right) - \left(\frac{x \cdot a}{t} + \frac{y \cdot z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{1}{\frac{a - t}{z - t}}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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))))