Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.7 → 2.0

Time: 12.5s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-156}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-230}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y - x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= x -4.5e-156)
   (+ x (/ (- y x) (/ t z)))
   (if (<= x 5.8e-230) (+ x (/ z (/ t (- y x)))) (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 (x <= -4.5e-156) {
		tmp = x + ((y - x) / (t / z));
	} else if (x <= 5.8e-230) {
		tmp = x + (z / (t / (y - x)));
	} else {
		tmp = fma((y - x), (z / t), x);
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -4.5e-156)
		tmp = Float64(x + Float64(Float64(y - x) / Float64(t / z)));
	elseif (x <= 5.8e-230)
		tmp = Float64(x + Float64(z / Float64(t / Float64(y - x))));
	else
		tmp = fma(Float64(y - x), Float64(z / t), x);
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_] := If[LessEqual[x, -4.5e-156], N[(x + N[(N[(y - x), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e-230], N[(x + N[(z / N[(t / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * N[(z / t), $MachinePrecision] + x), $MachinePrecision]]]

Target

Original	6.7
Target	2.0
Herbie	2.0

\[\begin{array}{l} \mathbf{if}\;x < -9.025511195533005 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x < 4.275032163700715 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if x < -4.49999999999999986e-156

   1. Initial program 7.2

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

   2. Simplified 0.9

      \[\leadsto \color{blue}{x + \frac{y - x}{\frac{t}{z}}} \]
      Proof

      [Start] 7.2	\[ x + \frac{\left(y - x\right) \cdot z}{t} \]
      associate-/l* [=>] 0.9	\[ x + \color{blue}{\frac{y - x}{\frac{t}{z}}} \]

   if -4.49999999999999986e-156 < x < 5.80000000000000011e-230

   1. Initial program 4.8

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

   2. Simplified 5.5

      \[\leadsto \color{blue}{x + \frac{y - x}{t} \cdot z} \]
      Proof

      [Start] 4.8	\[ x + \frac{\left(y - x\right) \cdot z}{t} \]
      associate-*l/ [<=] 5.5	\[ x + \color{blue}{\frac{y - x}{t} \cdot z} \]

   3. Applied egg-rr 5.4

      \[\leadsto x + \color{blue}{\frac{z}{\frac{t}{y - x}}} \]

   if 5.80000000000000011e-230 < x

   1. Initial program 7.1

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

   2. Simplified 1.3

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

      [Start] 7.1	\[ x + \frac{\left(y - x\right) \cdot z}{t} \]
      +-commutative [=>] 7.1	\[ \color{blue}{\frac{\left(y - x\right) \cdot z}{t} + x} \]
      associate-*r/ [<=] 1.3	\[ \color{blue}{\left(y - x\right) \cdot \frac{z}{t}} + x \]
      fma-def [=>] 1.3	\[ \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-156}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-230}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y - x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	1865

\[\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 10^{+307}\right):\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	10.7
Cost	1240

\[\begin{array}{l} t_1 := x - x \cdot \frac{z}{t}\\ t_2 := x + \frac{y \cdot z}{t}\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+278}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{-37}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -7.8 \cdot 10^{-174}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-255}:\\ \;\;\;\;x - z \cdot \frac{x}{t}\\ \mathbf{elif}\;y \leq 1.62 \cdot 10^{-103}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	27.7
Cost	1112

\[\begin{array}{l} t_1 := y \cdot \frac{z}{t}\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{-95}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-114}:\\ \;\;\;\;\frac{z}{\frac{t}{y}}\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{-88}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.32 \cdot 10^{+64}:\\ \;\;\;\;z \cdot \frac{-x}{t}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+72}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	24.2
Cost	1112

\[\begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{+47}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-13}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+115}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+149}:\\ \;\;\;\;\frac{z}{\frac{t}{y}}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+210}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+243}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	12.8
Cost	1108

\[\begin{array}{l} t_1 := x - z \cdot \frac{x}{t}\\ t_2 := x + \frac{y \cdot z}{t}\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{+279}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;y \leq -8 \cdot 10^{-37}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{-164}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-222}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	2.0
Cost	968

\[\begin{array}{l} \mathbf{if}\;x \leq -9.4 \cdot 10^{-157}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-216}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y - x}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{t}{z}}{y - x}}\\ \end{array} \]

Alternative 7
Error	26.2
Cost	849

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-30}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-110} \lor \neg \left(x \leq 1.26 \cdot 10^{-88}\right) \land x \leq 3.2 \cdot 10^{-13}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	26.5
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-90}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{-110}:\\ \;\;\;\;\frac{z}{\frac{t}{y}}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-88}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-14}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	4.6
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+169} \lor \neg \left(x \leq 2.9 \cdot 10^{+72}\right):\\ \;\;\;\;x - x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \end{array} \]

Alternative 10
Error	4.4
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+170} \lor \neg \left(x \leq 8.2 \cdot 10^{+72}\right):\\ \;\;\;\;x - x \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y - x}}\\ \end{array} \]

Alternative 11
Error	2.0
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-160} \lor \neg \left(x \leq 1.75 \cdot 10^{-240}\right):\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y - x}}\\ \end{array} \]

Alternative 12
Error	26.4
Cost	716

\[\begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-93}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-262}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-13}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	13.3
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{-147} \lor \neg \left(t \leq 2.45 \cdot 10^{-191}\right):\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \end{array} \]

Alternative 14
Error	31.7
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023039 
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< x -9.025511195533005e-135) (- x (* (/ z t) (- x y))) (if (< x 4.275032163700715e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

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