Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4

Average Error: 2.2 → 1.9

Time: 9.5s

Precision: binary64

Cost: 6980

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq 3.15 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y - x}{t}}{\frac{1}{z}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= z 3.15e-49)
   (fma (- y x) (/ z t) x)
   (+ x (/ (/ (- y x) t) (/ 1.0 z)))))

C

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

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.15e-49) {
		tmp = fma((y - x), (z / t), x);
	} else {
		tmp = x + (((y - x) / t) / (1.0 / z));
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 3.15e-49)
		tmp = fma(Float64(y - x), Float64(z / t), x);
	else
		tmp = Float64(x + Float64(Float64(Float64(y - x) / t) / Float64(1.0 / z)));
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_] := If[LessEqual[z, 3.15e-49], N[(N[(y - x), $MachinePrecision] * N[(z / t), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] / N[(1.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	2.2
Target	2.4
Herbie	1.9

\[\begin{array}{l} \mathbf{if}\;\left(y - x\right) \cdot \frac{z}{t} < -1013646692435.8867:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;\left(y - x\right) \cdot \frac{z}{t} < 0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if z < 3.1499999999999998e-49

   1. Initial program 1.8

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

   2. Simplified 1.8

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

      [Start] 1.8	\[ x + \left(y - x\right) \cdot \frac{z}{t} \]
      +-commutative [=>] 1.8	\[ \color{blue}{\left(y - x\right) \cdot \frac{z}{t} + x} \]
      fma-def [=>] 1.8	\[ \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)} \]

   if 3.1499999999999998e-49 < z

   1. Initial program 3.8

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

   2. Applied egg-rr 2.2

      \[\leadsto x + \color{blue}{\frac{\frac{y - x}{t}}{\frac{1}{z}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.15 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y - x}{t}}{\frac{1}{z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	16.0
Cost	1488

\[\begin{array}{l} t_1 := z \cdot \frac{y - x}{t}\\ \mathbf{if}\;\frac{z}{t} \leq -1 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{-167}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 10^{-112}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;\frac{z}{t} \leq 10^{-28}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	27.5
Cost	1379

\[\begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-168}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-270} \lor \neg \left(x \leq -2.8 \cdot 10^{-300}\right) \land \left(x \leq 4.1 \cdot 10^{-107} \lor \neg \left(x \leq 9.5 \cdot 10^{-96}\right) \land \left(x \leq 4.6 \cdot 10^{-66} \lor \neg \left(x \leq 1.65 \cdot 10^{+50}\right) \land x \leq 1.6 \cdot 10^{+86}\right)\right):\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	23.9
Cost	1360

\[\begin{array}{l} t_1 := \frac{y}{\frac{t}{z}}\\ \mathbf{if}\;\frac{z}{t} \leq -1 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{-167}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 10^{-28}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{t}{y}}\\ \end{array} \]

Alternative 4
Error	4.7
Cost	969

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -5 \lor \neg \left(\frac{z}{t} \leq 10^{-7}\right):\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 5
Error	4.7
Cost	969

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -5 \lor \neg \left(\frac{z}{t} \leq 10^{-7}\right):\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternative 6
Error	4.6
Cost	969

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -5 \lor \neg \left(\frac{z}{t} \leq 10^{-7}\right):\\ \;\;\;\;\frac{z}{\frac{t}{y - x}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternative 7
Error	1.6
Cost	836

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq 5 \cdot 10^{+146}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(y - x\right)}{t}\\ \end{array} \]

Alternative 8
Error	1.9
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \leq 3.8 \cdot 10^{-52}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{y - x}{t}}{\frac{1}{z}}\\ \end{array} \]

Alternative 9
Error	28.5
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+104} \lor \neg \left(y \leq 5.3 \cdot 10^{+70}\right):\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	31.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023060 
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"
  :precision binary64

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.8867) (+ x (/ (- y x) (/ t z))) (if (< (* (- y x) (/ z t)) 0.0) (+ x (/ (* (- y x) z) t)) (+ x (/ (- y x) (/ t z)))))

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