Numeric.Histogram:binBounds from Chart-1.5.3

Average Error: 6.6 → 1.6

Time: 9.8s

Precision: binary64

Cost: 1864

Math

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

↓

\[\begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+298}:\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+277}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y - x}}\\ \end{array} \]

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 -5e+298)
     (+ x (* z (/ (- y x) t)))
     (if (<= t_1 5e+277) t_1 (+ x (/ z (/ t (- y 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 <= -5e+298) {
		tmp = x + (z * ((y - x) / t));
	} else if (t_1 <= 5e+277) {
		tmp = t_1;
	} else {
		tmp = x + (z / (t / (y - x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + (((y - x) * z) / t)
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (((y - x) * z) / t)
    if (t_1 <= (-5d+298)) then
        tmp = x + (z * ((y - x) / t))
    else if (t_1 <= 5d+277) then
        tmp = t_1
    else
        tmp = x + (z / (t / (y - x)))
    end if
    code = tmp
end function

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 + (((y - x) * z) / t);
	double tmp;
	if (t_1 <= -5e+298) {
		tmp = x + (z * ((y - x) / t));
	} else if (t_1 <= 5e+277) {
		tmp = t_1;
	} else {
		tmp = x + (z / (t / (y - x)));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t):
	t_1 = x + (((y - x) * z) / t)
	tmp = 0
	if t_1 <= -5e+298:
		tmp = x + (z * ((y - x) / t))
	elif t_1 <= 5e+277:
		tmp = t_1
	else:
		tmp = x + (z / (t / (y - 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 <= -5e+298)
		tmp = Float64(x + Float64(z * Float64(Float64(y - x) / t)));
	elseif (t_1 <= 5e+277)
		tmp = t_1;
	else
		tmp = Float64(x + Float64(z / Float64(t / Float64(y - x))));
	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 + (((y - x) * z) / t);
	tmp = 0.0;
	if (t_1 <= -5e+298)
		tmp = x + (z * ((y - x) / t));
	elseif (t_1 <= 5e+277)
		tmp = t_1;
	else
		tmp = x + (z / (t / (y - x)));
	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[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+298], N[(x + N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+277], t$95$1, N[(x + N[(z / N[(t / N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	6.6
Target	1.9
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 (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < -5.0000000000000003e298

   1. Initial program 52.0

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

   2. Simplified 5.4

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

      [Start] 52.0	\[ x + \frac{\left(y - x\right) \cdot z}{t} \]
      associate-*l/ [<=] 5.4	\[ x + \color{blue}{\frac{y - x}{t} \cdot z} \]

   if -5.0000000000000003e298 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t)) < 4.99999999999999982e277

   1. Initial program 0.8

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

   if 4.99999999999999982e277 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) z) t))

   1. Initial program 40.9

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

   2. Simplified 9.0

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

      [Start] 40.9	\[ x + \frac{\left(y - x\right) \cdot z}{t} \]
      associate-*l/ [<=] 9.0	\[ x + \color{blue}{\frac{y - x}{t} \cdot z} \]

   3. Taylor expanded in y around 0 40.9

      \[\leadsto x + \color{blue}{\left(-1 \cdot \frac{z \cdot x}{t} + \frac{y \cdot z}{t}\right)} \]

   4. Simplified 8.6

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

      [Start] 40.9	\[ x + \left(-1 \cdot \frac{z \cdot x}{t} + \frac{y \cdot z}{t}\right) \]
      +-commutative [=>] 40.9	\[ x + \color{blue}{\left(\frac{y \cdot z}{t} + -1 \cdot \frac{z \cdot x}{t}\right)} \]
      associate-*l/ [<=] 26.3	\[ x + \left(\color{blue}{\frac{y}{t} \cdot z} + -1 \cdot \frac{z \cdot x}{t}\right) \]
      mul-1-neg [=>] 26.3	\[ x + \left(\frac{y}{t} \cdot z + \color{blue}{\left(-\frac{z \cdot x}{t}\right)}\right) \]
      associate-*r/ [<=] 9.0	\[ x + \left(\frac{y}{t} \cdot z + \left(-\color{blue}{z \cdot \frac{x}{t}}\right)\right) \]
      *-commutative [=>] 9.0	\[ x + \left(\frac{y}{t} \cdot z + \left(-\color{blue}{\frac{x}{t} \cdot z}\right)\right) \]
      distribute-lft-neg-out [<=] 9.0	\[ x + \left(\frac{y}{t} \cdot z + \color{blue}{\left(-\frac{x}{t}\right) \cdot z}\right) \]
      distribute-rgt-in [<=] 9.0	\[ x + \color{blue}{z \cdot \left(\frac{y}{t} + \left(-\frac{x}{t}\right)\right)} \]
      sub-neg [<=] 9.0	\[ x + z \cdot \color{blue}{\left(\frac{y}{t} - \frac{x}{t}\right)} \]
      div-sub [<=] 9.0	\[ x + z \cdot \color{blue}{\frac{y - x}{t}} \]
      associate-*r/ [=>] 40.9	\[ x + \color{blue}{\frac{z \cdot \left(y - x\right)}{t}} \]
      associate-/l* [=>] 8.6	\[ x + \color{blue}{\frac{z}{\frac{t}{y - x}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot z}{t} \leq -5 \cdot 10^{+298}:\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot z}{t} \leq 5 \cdot 10^{+277}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z}{\frac{t}{y - x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	21.2
Cost	1240

\[\begin{array}{l} t_1 := z \cdot \frac{y - x}{t}\\ t_2 := x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{if}\;y \leq -4.7 \cdot 10^{+160}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-141}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-10}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+197}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 2
Error	30.3
Cost	1113

\[\begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+158}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-241}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-183}:\\ \;\;\;\;\frac{-z}{\frac{t}{x}}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-11}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+56} \lor \neg \left(y \leq 5.2 \cdot 10^{+196}\right):\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	16.8
Cost	976

\[\begin{array}{l} t_1 := x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{-75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-182}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-199}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-202}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	26.4
Cost	849

\[\begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{-75}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-114} \lor \neg \left(x \leq -4.8 \cdot 10^{-199}\right) \land x \leq 2.1 \cdot 10^{-203}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	26.4
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-75}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-199}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-202}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	26.5
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-75}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-113}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-199}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-207}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	4.5
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -6.7 \cdot 10^{+89} \lor \neg \left(y \leq 7.2 \cdot 10^{+117}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y - x}{t}\\ \end{array} \]

Alternative 8
Error	17.8
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-226} \lor \neg \left(x \leq 9 \cdot 10^{-203}\right):\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 9
Error	8.3
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-57} \lor \neg \left(y \leq 3.7 \cdot 10^{-153}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \end{array} \]

Alternative 10
Error	9.6
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{-8} \lor \neg \left(y \leq 2.8 \cdot 10^{-157}\right):\\ \;\;\;\;x + y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{x}{t}\\ \end{array} \]

Alternative 11
Error	1.9
Cost	576

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

Alternative 12
Error	31.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022364 
(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)))