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

Average Error: 24.1 → 7.5

Time: 7.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y - x, \left(z - t\right) \cdot \frac{1}{a - t}, x\right)\\ \mathbf{elif}\;t_1 \leq 3.107901749914643 \cdot 10^{+254}:\\ \;\;\;\;\left(\frac{x \cdot t}{a - t} + \left(x + \frac{y \cdot z}{a - t}\right)\right) - \left(\frac{x \cdot z}{a - t} + \frac{y \cdot t}{a - t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z - t}{a - t}\\ \end{array} \]

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 (- INFINITY))
     (fma (- y x) (* (- z t) (/ 1.0 (- a t))) x)
     (if (<= t_1 3.107901749914643e+254)
       (-
        (+ (/ (* x t) (- a t)) (+ x (/ (* y z) (- a t))))
        (+ (/ (* x z) (- a t)) (/ (* y t) (- a t))))
       (+ x (* (- y x) (/ (- z t) (- a t))))))))

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 <= -((double) INFINITY)) {
		tmp = fma((y - x), ((z - t) * (1.0 / (a - t))), x);
	} else if (t_1 <= 3.107901749914643e+254) {
		tmp = (((x * t) / (a - t)) + (x + ((y * z) / (a - t)))) - (((x * z) / (a - t)) + ((y * t) / (a - t)));
	} else {
		tmp = x + ((y - x) * ((z - t) / (a - t)));
	}
	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 <= Float64(-Inf))
		tmp = fma(Float64(y - x), Float64(Float64(z - t) * Float64(1.0 / Float64(a - t))), x);
	elseif (t_1 <= 3.107901749914643e+254)
		tmp = Float64(Float64(Float64(Float64(x * t) / Float64(a - t)) + Float64(x + Float64(Float64(y * z) / Float64(a - t)))) - Float64(Float64(Float64(x * z) / Float64(a - t)) + Float64(Float64(y * t) / Float64(a - t))));
	else
		tmp = Float64(x + Float64(Float64(y - x) * Float64(Float64(z - t) / Float64(a - t))));
	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, (-Infinity)], N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] * N[(1.0 / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 3.107901749914643e+254], N[(N[(N[(N[(x * t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + N[(x + N[(N[(y * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(x * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + N[(N[(y * t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.1
Target	9.2
Herbie	7.5

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.0

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

   2. Simplified 18.0

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

   3. Applied egg-rr 18.1

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

   if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 3.10790174991464297e254

   1. Initial program 8.7

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

   2. Simplified 9.2

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

   3. Taylor expanded in y around 0 3.5

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

   if 3.10790174991464297e254 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 55.4

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

   2. Simplified 15.7

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

   3. Applied egg-rr 15.8

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

   4. Applied egg-rr 16.4

      \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{\frac{z - t}{{\left(\sqrt[3]{a - t}\right)}^{2}}}{\sqrt[3]{a - t}}}, x\right) \]

   5. Applied egg-rr 15.7

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

4. Final simplification 7.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y - x, \left(z - t\right) \cdot \frac{1}{a - t}, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 3.107901749914643 \cdot 10^{+254}:\\ \;\;\;\;\left(\frac{x \cdot t}{a - t} + \left(x + \frac{y \cdot z}{a - t}\right)\right) - \left(\frac{x \cdot z}{a - t} + \frac{y \cdot t}{a - t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z - t}{a - t}\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(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))))