Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.5 → 0.6

Time: 10.2s

Precision: binary64

Cost: 3152

Math

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

↓

\[\begin{array}{l} t_1 := x + z \cdot \frac{y - x}{t}\\ t_2 := x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{if}\;t_2 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{-80}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-180}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 \leq 10^{+305}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (* z (/ (- y x) t)))) (t_2 (+ x (/ (* (- y x) z) t))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -2e-80)
       t_2
       (if (<= t_2 2e-180) t_1 (if (<= t_2 1e+305) t_2 t_1))))))

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 + (z * ((y - x) / t));
	double t_2 = x + (((y - x) * z) / t);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -2e-80) {
		tmp = t_2;
	} else if (t_2 <= 2e-180) {
		tmp = t_1;
	} else if (t_2 <= 1e+305) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

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

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = x + (z * ((y - x) / t));
	double t_2 = x + (((y - x) * z) / t);
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= -2e-80) {
		tmp = t_2;
	} else if (t_2 <= 2e-180) {
		tmp = t_1;
	} else if (t_2 <= 1e+305) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return x + (((y - x) * z) / t)

↓

def code(x, y, z, t):
	t_1 = x + (z * ((y - x) / t))
	t_2 = x + (((y - x) * z) / t)
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= -2e-80:
		tmp = t_2
	elif t_2 <= 2e-180:
		tmp = t_1
	elif t_2 <= 1e+305:
		tmp = t_2
	else:
		tmp = t_1
	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(z * Float64(Float64(y - x) / t)))
	t_2 = Float64(x + Float64(Float64(Float64(y - x) * z) / t))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -2e-80)
		tmp = t_2;
	elseif (t_2 <= 2e-180)
		tmp = t_1;
	elseif (t_2 <= 1e+305)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + (((y - x) * z) / t);
end

↓

function tmp_2 = code(x, y, z, t)
	t_1 = x + (z * ((y - x) / t));
	t_2 = x + (((y - x) * z) / t);
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= -2e-80)
		tmp = t_2;
	elseif (t_2 <= 2e-180)
		tmp = t_1;
	elseif (t_2 <= 1e+305)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = 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[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -2e-80], t$95$2, If[LessEqual[t$95$2, 2e-180], t$95$1, If[LessEqual[t$95$2, 1e+305], t$95$2, t$95$1]]]]]]

Target

Original	6.5
Target	2.0
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < -inf.0 or -1.99999999999999992e-80 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < 2e-180 or 9.9999999999999994e304 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t))

   1. Initial program 27.4

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

   2. Simplified 1.8

      \[\leadsto \color{blue}{x + \frac{y - x}{t} \cdot z} \]
      Proof
      (+.f64 x (*.f64 (/.f64 (-.f64 y x) t) z)): 0 points increase in error, 0 points decrease in error
      (+.f64 x (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (-.f64 y x) z) t))): 38 points increase in error, 40 points decrease in error

   if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < -1.99999999999999992e-80 or 2e-180 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < 9.9999999999999994e304

   1. Initial program 0.3

      \[x + \frac{\left(y - x\right) \cdot z}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} \leq -\infty:\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \leq -2 \cdot 10^{-80}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \leq 2 \cdot 10^{-180}:\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \leq 10^{+305}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.3
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-278}:\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \]

Alternative 2
Error	11.9
Cost	1636

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{t}{z}}\\ t_2 := z \cdot \frac{y - x}{t}\\ t_3 := x - x \cdot \frac{z}{t}\\ \mathbf{if}\;x \leq -4.3 \cdot 10^{+88}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5.6 \cdot 10^{-90}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-140}:\\ \;\;\;\;x - z \cdot \frac{x}{t}\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-159}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-12}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y}}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+20}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;x \leq 10^{+50}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	12.3
Cost	1636

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{t}{z}}\\ t_2 := \frac{\left(y - x\right) \cdot z}{t}\\ t_3 := x - x \cdot \frac{z}{t}\\ t_4 := x + \frac{z}{\frac{t}{y}}\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{+88}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8.8 \cdot 10^{-110}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-131}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-140}:\\ \;\;\;\;x - z \cdot \frac{x}{t}\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-159}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-10}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+25}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+50}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	27.6
Cost	1244

\[\begin{array}{l} t_1 := z \cdot \frac{x}{-t}\\ t_2 := z \cdot \frac{y}{t}\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{-36}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-131}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-227}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-257}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-128}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	27.7
Cost	1244

\[\begin{array}{l} t_1 := z \cdot \frac{y}{t}\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{-36}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-89}:\\ \;\;\;\;z \cdot \frac{x}{-t}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-131}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-159}:\\ \;\;\;\;\frac{x \cdot \left(-z\right)}{t}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-227}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-257}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-125}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	12.4
Cost	1240

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{t}{z}}\\ t_2 := x - x \cdot \frac{z}{t}\\ \mathbf{if}\;x \leq -4.3 \cdot 10^{+88}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-159}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y}}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+20}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+50}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	22.3
Cost	976

\[\begin{array}{l} t_1 := z \cdot \frac{y - x}{t}\\ \mathbf{if}\;z \leq -3.1 \cdot 10^{-38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.6 \cdot 10^{-116}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -3.15 \cdot 10^{-140}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;z \leq 270000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	4.5
Cost	840

\[\begin{array}{l} t_1 := x - x \cdot \frac{z}{t}\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{+89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+107}:\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	2.2
Cost	840

\[\begin{array}{l} t_1 := x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-276}:\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	21.9
Cost	712

\[\begin{array}{l} t_1 := z \cdot \frac{y - x}{t}\\ \mathbf{if}\;z \leq -1.42 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 19500:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	10.8
Cost	712

\[\begin{array}{l} t_1 := x + \frac{y}{\frac{t}{z}}\\ \mathbf{if}\;t \leq -5 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-147}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	11.0
Cost	712

\[\begin{array}{l} t_1 := x + \frac{z}{\frac{t}{y}}\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-126}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	28.6
Cost	584

\[\begin{array}{l} t_1 := y \cdot \frac{z}{t}\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+27}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	28.4
Cost	584

\[\begin{array}{l} t_1 := z \cdot \frac{y}{t}\\ \mathbf{if}\;z \leq -5.4 \cdot 10^{-35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+27}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	28.4
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-35}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+28}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternative 16
Error	31.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022330 
(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)))