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

Average Error: 24.5 → 6.5

Time: 5.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+286}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a - t}, x\right)\\ \mathbf{elif}\;t_1 \leq -5 \cdot 10^{-305}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\left(y + a \cdot \frac{y}{t}\right) + \left(\frac{x}{\frac{t}{z}} - \left(x \cdot \frac{a}{t} + z \cdot \frac{y}{t}\right)\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\left(\frac{x \cdot t}{a - t} + \left(x + \frac{y \cdot z}{a - t}\right)\right) - \left(\frac{x \cdot z}{a - t} + \frac{y \cdot t}{a - t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - t, {\left(\frac{a - t}{y - x}\right)}^{-1}, x\right)\\ \end{array} \]

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 -5e+286)
     (fma (- z t) (/ (- y x) (- a t)) x)
     (if (<= t_1 -5e-305)
       t_1
       (if (<= t_1 0.0)
         (+
          (+ y (* a (/ y t)))
          (- (/ x (/ t z)) (+ (* x (/ a t)) (* z (/ y t)))))
         (if (<= t_1 5e+298)
           (-
            (+ (/ (* x t) (- a t)) (+ x (/ (* y z) (- a t))))
            (+ (/ (* x z) (- a t)) (/ (* y t) (- a t))))
           (fma (- z t) (pow (/ (- a t) (- y x)) -1.0) 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 <= -5e+286) {
		tmp = fma((z - t), ((y - x) / (a - t)), x);
	} else if (t_1 <= -5e-305) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (y + (a * (y / t))) + ((x / (t / z)) - ((x * (a / t)) + (z * (y / t))));
	} else if (t_1 <= 5e+298) {
		tmp = (((x * t) / (a - t)) + (x + ((y * z) / (a - t)))) - (((x * z) / (a - t)) + ((y * t) / (a - t)));
	} else {
		tmp = fma((z - t), pow(((a - t) / (y - x)), -1.0), 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 <= -5e+286)
		tmp = fma(Float64(z - t), Float64(Float64(y - x) / Float64(a - t)), x);
	elseif (t_1 <= -5e-305)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(y + Float64(a * Float64(y / t))) + Float64(Float64(x / Float64(t / z)) - Float64(Float64(x * Float64(a / t)) + Float64(z * Float64(y / t)))));
	elseif (t_1 <= 5e+298)
		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 = fma(Float64(z - t), (Float64(Float64(a - t) / Float64(y - x)) ^ -1.0), 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, -5e+286], N[(N[(z - t), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, -5e-305], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(y + N[(a * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x / N[(t / z), $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[(a / t), $MachinePrecision]), $MachinePrecision] + N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+298], 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[(N[(z - t), $MachinePrecision] * N[Power[N[(N[(a - t), $MachinePrecision] / N[(y - x), $MachinePrecision]), $MachinePrecision], -1.0], $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	24.5
Target	9.5
Herbie	6.5

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

Derivation

1. Split input into 5 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -5.0000000000000004e286

   1. Initial program 60.0

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

   2. Simplified 18.3

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

   3. Applied egg-rr 18.3

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

   if -5.0000000000000004e286 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -4.99999999999999985e-305

   1. Initial program 1.9

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

   if -4.99999999999999985e-305 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 61.3

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

   2. Simplified 61.3

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

   3. Taylor expanded in t around inf 0.2

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

   4. Simplified 0.8

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

   if 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 5.0000000000000003e298

   1. Initial program 1.6

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

   2. Simplified 6.8

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

   3. Taylor expanded in z around 0 1.3

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

   if 5.0000000000000003e298 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 62.0

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

   2. Simplified 17.5

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

   3. Applied egg-rr 17.6

      \[\leadsto \mathsf{fma}\left(z - t, \color{blue}{{\left(\frac{a - t}{y - x}\right)}^{-1}}, x\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 6.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -5 \cdot 10^{+286}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a - t}, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -5 \cdot 10^{-305}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 0:\\ \;\;\;\;\left(y + a \cdot \frac{y}{t}\right) + \left(\frac{x}{\frac{t}{z}} - \left(x \cdot \frac{a}{t} + z \cdot \frac{y}{t}\right)\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\left(\frac{x \cdot t}{a - t} + \left(x + \frac{y \cdot z}{a - t}\right)\right) - \left(\frac{x \cdot z}{a - t} + \frac{y \cdot t}{a - t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - t, {\left(\frac{a - t}{y - x}\right)}^{-1}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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))))