Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, A

Average Error: 7.6 → 2.3

Time: 16.4s

Precision: binary64

Cost: 3401

Math

\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

↓

\[\begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t_1}}{x + 1}\\ \mathbf{if}\;t_2 \leq -\infty \lor \neg \left(t_2 \leq 10^{+27}\right):\\ \;\;\;\;\frac{x + \frac{y}{\frac{t_1}{z}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x)) (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (or (<= t_2 (- INFINITY)) (not (<= t_2 1e+27)))
     (/ (+ x (/ y (/ t_1 z))) (+ x 1.0))
     t_2)))

C

double code(double x, double y, double z, double t) {
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
}

↓

double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if ((t_2 <= -((double) INFINITY)) || !(t_2 <= 1e+27)) {
		tmp = (x + (y / (t_1 / z))) / (x + 1.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
}

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if ((t_2 <= -Double.POSITIVE_INFINITY) || !(t_2 <= 1e+27)) {
		tmp = (x + (y / (t_1 / z))) / (x + 1.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)

↓

def code(x, y, z, t):
	t_1 = (z * t) - x
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0)
	tmp = 0
	if (t_2 <= -math.inf) or not (t_2 <= 1e+27):
		tmp = (x + (y / (t_1 / z))) / (x + 1.0)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	return Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / Float64(Float64(t * z) - x))) / Float64(x + 1.0))
end

↓

function code(x, y, z, t)
	t_1 = Float64(Float64(z * t) - x)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	tmp = 0.0
	if ((t_2 <= Float64(-Inf)) || !(t_2 <= 1e+27))
		tmp = Float64(Float64(x + Float64(y / Float64(t_1 / z))) / Float64(x + 1.0));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0);
end

↓

function tmp_2 = code(x, y, z, t)
	t_1 = (z * t) - x;
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	tmp = 0.0;
	if ((t_2 <= -Inf) || ~((t_2 <= 1e+27)))
		tmp = (x + (y / (t_1 / z))) / (x + 1.0);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / N[(N[(t * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$2, (-Infinity)], N[Not[LessEqual[t$95$2, 1e+27]], $MachinePrecision]], N[(N[(x + N[(y / N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$2]]]

Target

Original	7.6
Target	0.3
Herbie	2.3

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < -inf.0 or 1e27 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1))

   1. Initial program 37.4

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]

   2. Simplified 37.4

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

      [Start] 37.4	\[ \frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
      *-commutative [=>] 37.4	\[ \frac{x + \frac{y \cdot z - x}{\color{blue}{z \cdot t} - x}}{x + 1} \]

   3. Taylor expanded in y around inf 37.4

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

   4. Simplified 9.0

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

      [Start] 37.4	\[ \frac{x + \frac{y \cdot z}{t \cdot z - x}}{x + 1} \]
      associate-/l* [=>] 9.0	\[ \frac{x + \color{blue}{\frac{y}{\frac{t \cdot z - x}{z}}}}{x + 1} \]

   if -inf.0 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < 1e27

   1. Initial program 0.8

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -\infty \lor \neg \left(\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+27}\right):\\ \;\;\;\;\frac{x + \frac{y}{\frac{z \cdot t - x}{z}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \end{array} \]

Alternatives

Alternative 1
Error	17.3
Cost	1360

\[\begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := \frac{x + \frac{x - y \cdot z}{x}}{x + 1}\\ \mathbf{if}\;y \leq -2.65 \cdot 10^{+251}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{-85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-16}:\\ \;\;\;\;\frac{x - \frac{x}{z \cdot t - x}}{x + 1}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+227}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	14.2
Cost	1232

\[\begin{array}{l} t_1 := 1 - y \cdot \frac{z}{x \cdot x}\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{+56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-54}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{x - \frac{x}{z \cdot t - x}}{x + 1}\\ \mathbf{elif}\;x \leq 0.00066:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{t - \frac{x}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	7.3
Cost	1225

\[\begin{array}{l} t_1 := z \cdot t - x\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{-86} \lor \neg \left(y \leq 1.75 \cdot 10^{-65}\right):\\ \;\;\;\;\frac{x + \frac{y}{\frac{t_1}{z}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \frac{x}{t_1}}{x + 1}\\ \end{array} \]

Alternative 4
Error	7.0
Cost	1224

\[\begin{array}{l} t_1 := z \cdot t - x\\ \mathbf{if}\;y \leq -6.3 \cdot 10^{-86}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t}{y} - \frac{x}{y \cdot z}}}{x + 1}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-66}:\\ \;\;\;\;\frac{x - \frac{x}{t_1}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{\frac{t_1}{z}}}{x + 1}\\ \end{array} \]

Alternative 5
Error	20.7
Cost	976

\[\begin{array}{l} t_1 := \frac{x}{x + 1}\\ t_2 := \frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{if}\;x \leq -5.2 \cdot 10^{-134}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-144}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-10}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	20.7
Cost	976

\[\begin{array}{l} t_1 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -8.4 \cdot 10^{-134}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.3 \cdot 10^{-145}:\\ \;\;\;\;\frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{y}{t + x \cdot t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	14.1
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+56} \lor \neg \left(x \leq 0.00095\right):\\ \;\;\;\;1 - y \cdot \frac{z}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]

Alternative 8
Error	14.0
Cost	840

\[\begin{array}{l} \mathbf{if}\;x \leq -45000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.00095:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	20.8
Cost	588

\[\begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-15}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-134}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.45 \cdot 10^{-130}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	20.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{-133}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-127}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	26.8
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-16}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-52}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Error	28.7
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023060 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1.0))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))