Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 10.38% → 1.91%

Time: 10.2s

Precision: binary64

Cost: 8136

Math

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

↓

\[\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+251}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;t_1 \leq 10^{+255}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y x) z) t))))
   (if (<= t_1 -2e+251)
     (+ x (/ (- y x) (/ t z)))
     (if (<= t_1 1e+255) t_1 (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 t_1 = x + (((y - x) * z) / t);
	double tmp;
	if (t_1 <= -2e+251) {
		tmp = x + ((y - x) / (t / z));
	} else if (t_1 <= 1e+255) {
		tmp = t_1;
	} 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)
	t_1 = Float64(x + Float64(Float64(Float64(y - x) * z) / t))
	tmp = 0.0
	if (t_1 <= -2e+251)
		tmp = Float64(x + Float64(Float64(y - x) / Float64(t / z)));
	elseif (t_1 <= 1e+255)
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+251], N[(x + N[(N[(y - x), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+255], t$95$1, N[(N[(y - x), $MachinePrecision] * N[(z / t), $MachinePrecision] + x), $MachinePrecision]]]]

Target

Original	10.38%
Target	3.3%
Herbie	1.91%

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < -2.0000000000000001e251

   1. Initial program 45.67

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

   2. Simplified 4.85

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

      [Start] 45.67	\[ x + \frac{\left(y - x\right) \cdot z}{t} \]
      associate-/l* [=>] 4.85	\[ x + \color{blue}{\frac{y - x}{\frac{t}{z}}} \]

   if -2.0000000000000001e251 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < 9.99999999999999988e254

   1. Initial program 1.22

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

   if 9.99999999999999988e254 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t))

   1. Initial program 48.64

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

   2. Simplified 4.56

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

      [Start] 48.64	\[ x + \frac{\left(y - x\right) \cdot z}{t} \]
      +-commutative [=>] 48.64	\[ \color{blue}{\frac{\left(y - x\right) \cdot z}{t} + x} \]
      associate-*r/ [<=] 4.56	\[ \color{blue}{\left(y - x\right) \cdot \frac{z}{t}} + x \]
      fma-def [=>] 4.56	\[ \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.91

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} \leq -2 \cdot 10^{+251}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \leq 10^{+255}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.32%
Cost	1865

\[\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 10^{+267}\right):\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	1.77%
Cost	1865

\[\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+251} \lor \neg \left(t_1 \leq 2 \cdot 10^{+277}\right):\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	41.02%
Cost	980

\[\begin{array}{l} t_1 := \frac{y}{\frac{t}{z}}\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{-48}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.25 \cdot 10^{-166}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8.8 \cdot 10^{-187}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-278}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-105}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	36.7%
Cost	977

\[\begin{array}{l} \mathbf{if}\;t \leq -2250000:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-99} \lor \neg \left(t \leq 3.65 \cdot 10^{-66}\right) \land t \leq 3.5 \cdot 10^{+101}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	6.54%
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{-51} \lor \neg \left(z \leq 4.2 \cdot 10^{-119}\right):\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 6
Error	19.08%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -52000 \lor \neg \left(t \leq 4.6 \cdot 10^{-219}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \end{array} \]

Alternative 7
Error	14.73%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-137} \lor \neg \left(y \leq 1.86 \cdot 10^{-211}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{x}{t}\\ \end{array} \]

Alternative 8
Error	14.14%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-129} \lor \neg \left(y \leq 3.3 \cdot 10^{-208}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot \frac{z}{t}\\ \end{array} \]

Alternative 9
Error	18.69%
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -130000:\\ \;\;\;\;x + z \cdot \frac{y}{t}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-206}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 10
Error	18.78%
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -52000:\\ \;\;\;\;x + \frac{z}{\frac{t}{y}}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-212}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 11
Error	40.91%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-41}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-105}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	40.41%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-38}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-107}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	50.19%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023088 
(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)))