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

Average Error: 2.3 → 1.5

Time: 1.7min

Precision: binary64

Cost: 7108

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -5 \cdot 10^{+280}:\\ \;\;\;\;\frac{z \cdot \left(y - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ z t) -5e+280) (/ (* z (- y x)) t) (fma (- y x) (/ z t) x)))

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 / t) <= -5e+280) {
		tmp = (z * (y - x)) / t;
	} else {
		tmp = fma((y - x), (z / t), x);
	}
	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 (Float64(z / t) <= -5e+280)
		tmp = Float64(Float64(z * Float64(y - x)) / t);
	else
		tmp = fma(Float64(y - x), Float64(z / t), x);
	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[N[(z / t), $MachinePrecision], -5e+280], N[(N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * N[(z / t), $MachinePrecision] + x), $MachinePrecision]]

Target

Original	2.3
Target	2.5
Herbie	1.5

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 z t) < -5.0000000000000002e280

   1. Initial program 47.3

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

   2. Simplified 47.3

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

   3. Taylor expanded in z around -inf 0.3

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

   if -5.0000000000000002e280 < (/.f64 z t)

   1. Initial program 1.5

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

   2. Simplified 1.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)} \]
      Proof
3. Recombined 2 regimes into one program.

Alternatives

Alternative 1
Error	21.9
Cost	1684

\[\begin{array}{l} t_1 := \frac{-z}{t} \cdot x\\ t_2 := \frac{z}{t} \cdot y\\ \mathbf{if}\;\frac{z}{t} \leq -5 \cdot 10^{+190}:\\ \;\;\;\;\frac{y}{t} \cdot z\\ \mathbf{elif}\;\frac{z}{t} \leq -50000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq -5 \cdot 10^{-59}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 10^{+151}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	21.7
Cost	1684

\[\begin{array}{l} t_1 := \frac{z}{t} \cdot y\\ \mathbf{if}\;\frac{z}{t} \leq -5 \cdot 10^{+190}:\\ \;\;\;\;\frac{y}{t} \cdot z\\ \mathbf{elif}\;\frac{z}{t} \leq -50000000000:\\ \;\;\;\;\frac{-z}{t} \cdot x\\ \mathbf{elif}\;\frac{z}{t} \leq -5 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot z}{-t}\\ \end{array} \]

Alternative 3
Error	22.1
Cost	1164

\[\begin{array}{l} t_1 := \frac{z}{t} \cdot y\\ \mathbf{if}\;\frac{z}{t} \leq -5 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-t} \cdot z\\ \end{array} \]

Alternative 4
Error	14.9
Cost	968

\[\begin{array}{l} t_1 := z \cdot \frac{y - x}{t}\\ \mathbf{if}\;\frac{z}{t} \leq -2 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{-11}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	5.8
Cost	968

\[\begin{array}{l} t_1 := z \cdot \frac{y - x}{t}\\ \mathbf{if}\;\frac{z}{t} \leq -200:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 0.002:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	6.1
Cost	968

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\frac{z \cdot \left(y - x\right)}{t}\\ \mathbf{elif}\;\frac{z}{t} \leq 0.002:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \end{array} \]

Alternative 7
Error	22.4
Cost	840

\[\begin{array}{l} t_1 := \frac{z}{t} \cdot y\\ \mathbf{if}\;\frac{z}{t} \leq -5 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{-25}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	1.5
Cost	836

\[\begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -5 \cdot 10^{+280}:\\ \;\;\;\;\frac{z \cdot \left(y - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z}{t}\\ \end{array} \]

Alternative 9
Error	16.1
Cost	712

\[\begin{array}{l} t_1 := \frac{t - z}{t} \cdot x\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{-111}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	26.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-121}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-121}:\\ \;\;\;\;\frac{y}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	31.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023033 
(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))))