Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.5 → 1.6

Time: 11.8s

Precision: binary64

Cost: 7112

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{+24}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-123}:\\ \;\;\;\;x + \frac{\frac{z}{\frac{1}{y - x}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= t -7.8e+24)
   (+ x (/ (- y x) (/ t z)))
   (if (<= t 9.5e-123)
     (+ x (/ (/ z (/ 1.0 (- 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 (t <= -7.8e+24) {
		tmp = x + ((y - x) / (t / z));
	} else if (t <= 9.5e-123) {
		tmp = x + ((z / (1.0 / (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(Float64(y - x) * z) / t))
end

↓

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -7.8e+24)
		tmp = Float64(x + Float64(Float64(y - x) / Float64(t / z)));
	elseif (t <= 9.5e-123)
		tmp = Float64(x + Float64(Float64(z / Float64(1.0 / 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[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := If[LessEqual[t, -7.8e+24], N[(x + N[(N[(y - x), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.5e-123], N[(x + N[(N[(z / N[(1.0 / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(y - x), $MachinePrecision] * N[(z / t), $MachinePrecision] + x), $MachinePrecision]]]

Target

Original	6.5
Target	2.0
Herbie	1.6

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

Derivation

1. Split input into 3 regimes

2. if t < -7.7999999999999995e24

   1. Initial program 10.0

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

   2. Simplified 1.4

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

      [Start] 10.0	\[ x + \frac{\left(y - x\right) \cdot z}{t} \]
      associate-/l* [=>] 1.4	\[ x + \color{blue}{\frac{y - x}{\frac{t}{z}}} \]

   if -7.7999999999999995e24 < t < 9.5000000000000002e-123

   1. Initial program 2.2

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

   2. Applied egg-rr 2.2

      \[\leadsto x + \frac{\color{blue}{\frac{z}{\frac{1}{y - x}}}}{t} \]

   if 9.5000000000000002e-123 < t

   1. Initial program 7.3

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

   2. Simplified 1.2

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

      [Start] 7.3	\[ x + \frac{\left(y - x\right) \cdot z}{t} \]
      +-commutative [=>] 7.3	\[ \color{blue}{\frac{\left(y - x\right) \cdot z}{t} + x} \]
      associate-*r/ [<=] 1.2	\[ \color{blue}{\left(y - x\right) \cdot \frac{z}{t}} + x \]
      fma-def [=>] 1.2	\[ \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{+24}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-123}:\\ \;\;\;\;x + \frac{\frac{z}{\frac{1}{y - x}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	1865

\[\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 2 \cdot 10^{+303}\right):\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.8
Cost	1864

\[\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+283}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \]

Alternative 3
Error	29.1
Cost	1376

\[\begin{array}{l} t_1 := \frac{-x}{\frac{t}{z}}\\ \mathbf{if}\;x \leq -6.6 \cdot 10^{+111}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.35 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-90}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.25 \cdot 10^{-93}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{-36}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+21}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{+96}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	29.1
Cost	1376

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+111}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{+35}:\\ \;\;\;\;\frac{-x}{\frac{t}{z}}\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-89}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-88}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-33}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+19}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{+79}:\\ \;\;\;\;\frac{x \cdot \left(-z\right)}{t}\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{+96}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	29.1
Cost	1376

\[\begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+111}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{+35}:\\ \;\;\;\;\frac{-x}{\frac{t}{z}}\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-89}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-93}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-35}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+20}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+76}:\\ \;\;\;\;\frac{z}{t} \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{+96}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	17.9
Cost	977

\[\begin{array}{l} t_1 := x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{if}\;x \leq -8.6 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-235}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-220} \lor \neg \left(x \leq 2.7 \cdot 10^{-93}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternative 7
Error	18.8
Cost	976

\[\begin{array}{l} t_1 := x \cdot \left(1 - \frac{z}{t}\right)\\ t_2 := z \cdot \frac{y - x}{t}\\ \mathbf{if}\;z \leq -4.3 \cdot 10^{-8}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	29.2
Cost	850

\[\begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-14} \lor \neg \left(z \leq 5.4 \cdot 10^{-104} \lor \neg \left(z \leq 3.1 \cdot 10^{-86}\right) \land z \leq 31000000000000\right):\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	29.0
Cost	849

\[\begin{array}{l} \mathbf{if}\;z \leq -2.65 \cdot 10^{-14}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-106}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 10^{-84} \lor \neg \left(z \leq 3.35 \cdot 10^{+16}\right):\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	29.0
Cost	848

\[\begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-14}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-104}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+15}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 11
Error	4.1
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{-148} \lor \neg \left(z \leq 2 \cdot 10^{-51}\right):\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z}{t}\\ \end{array} \]

Alternative 12
Error	9.1
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{+34} \lor \neg \left(x \leq 3.7 \cdot 10^{+19}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 13
Error	9.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+35}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+20}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot \frac{z}{t}\\ \end{array} \]

Alternative 14
Error	31.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023012 
(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)))